
COPYRIGHT DEPOSm 



a 



WORKS OF 
PROF. WALTER LORING WEBB 

PUBLISHED BY 

JOHN WILEY & SONS. 



Railroad Construction. — Theory and Practice. 

A Text-book for the Use of Students in Colleges 
and Technical Schools. Fifth Edition. Revised 
and Enlarged. 16mo. xvii + 789 pages and 217 
figures and 10 plates. Morocco, $5.00. 

Problems In the Use and Adjustment of Ensrlneer- 
ins: Instruments. 

Forms for Field-notes; General Instructions for 
Extended Students' Surveys. 16mo. Morocco, 
S1.25. 

The Economics of Railroad Construction. 

Small 8vo. Second Edition, vii -j-347 pages, 35 
figures. Cloth, $2.50. 

The American Civil Engineers' Pocket Book 

(Author of Section on Railroads.) 
Large 16mo. Morocco, $5.00. 



^1 



MILEOAD CONSTEUCTION. 



THEOEY AND PEACTICE. 



A TEXT-BOOK FOB THE USE OF STUDENTS 
IN COLLEGES AND TECHNICAL SCHOOLS. 



BY 

WALTEE LOEING WEBB, O.E., 

Member American Society of Civil Engineers; Member American Railvmy 

Engineering Association; Assistant Professor of Civil Engineer- 

ing {Railroad Engineering) in the University of 

Pennsylvania^ 1893-1901; etc. 



FIFTH EDITION, REVISED AND ENLARGED. 

TOTAL ISSUE ELEVEN THOUSAND. 



NEW YOEK: 
JOHN WILEY & SONS. 
London: CHAPMAN & HALL, Limited. 
1913 



/ 






Copyright, 1899, 1903, 1908, 1913, 

BY 

WALTER LORING WEBB. 



t 



©Ci,A34382; 



& 



PEEFACE TO FIRST EDITION. 



The preparation of this book was begun several years ago, 
when much of the subject-matter treated was not to be found in 
print, or was scattered through many books and pamphlets, and 
was hence unavailable for student use. Portions of the book 
have already been printed by the mimeograph process or have 
been used as lecture-notes, and hence have been subjected to 
the refining process of class-room use. 

The author would call special attention to the follo\\ing 
features : 

a. Transition curves; the multiform-compound-curve method 
is used, which has been followed by many railroads in this 
country; the particular curves here developed have the great 
advantage of being exceedingly simple, and although the method 
is not theoretically exact, iris demonstrable that ,the differences 
are so small that they may safely be neglected. 

h. A system of earthwork computations which makes the multi- 
plication and reduction in a single operation by means of a slide- 
rule and which enables one to compute readily the volume of the 
most complicated earthwork forms with an accuracy which ic only 
limited by the precision of the cross-sectioning. 

c. The ''mass curve" in earthwork; the theory and use of 
this very valuable process. 

d. Tables I, II, III, and IV have been computed ah novo. 
Tables I and II were checked (after computation) mth other 
tables, which are generally considered as standard, and all 
discrepancies were further examined. They are believed to be 
perfect. 

e. Tables V, VI, VII, and IX have been borrowed, by per- 
mission, from ''Ludlow's Mathematical Tables.'' It is believed 
that five-place tables give as accurate results as actual field 



IV PREFACE TO SECOND EDITION, 

practice requires. Tables VIII and X have been compiled to 
conform with Ludlow's system. 

The author wishes to acknowledge his indebtedness to Mr, 
Chas. A. Sims, civil engineer and railroad contractor, for reading 
and revising the portions relating to the cost of earthwork. 

Since the book is written primarily for students of railroad 
engineering in technical institutions, the author has assumed 
the usual previous preparation in algebra, geometry, and trigo- 
nometry. 

Walter Loring Webb. 
University of Pennsylvania, 
Philadelphia, 
Jan. 1 . 1900. 



PREFACE TO SECOND EDITION. 



Since the issue of the first edition the author has conferred 
with many noted educators in civil engineering, among them the 
late Professors E. A. Fuertes and J. B. Johnson, regarding the 
most desirable size of page for this book. The inconvenience of 
the octavo edition for field-work was found to be limiting its use. 
It was therefore decided to recast the whole work and reduce 
the page from '^ octavo'' to ^^pocket-book" size. Advantage 
was then taken of the opportunity to revise freely and to add 
new matter. The original text has now been almost doubled by 
the addition of several chapters on structures, train resistance, 
rolling stock, etc., and also several chapters giving the funda- 
mental principles of the economics of railroad location. Those 
who are familiar with the late Mr. Wellington's masterpiece, 
''The Economic Theory of Railway Location," will readily ap- 
preciate the author's indebtedness to that work. But w^hile the 
same general method has been followed, the author has taken 
advantage of the classification of operating expenses adopted 
by the Interstate Commerce Commission, has used the figures 



PREFACE TO THIRD AND FOURTH EDITION^ V 

published by them (Avhich were unavailable when Mr. Welling- 
ton wrote), and has developed the theory on an independent 
basis, T\'ith the exception of a few minor details. Those who 
deny the utility of such methods of computation are referred to 
§§ 367, 426, and elsewhere for a practical discussion of that 
subject. 

The author's primary aim has been to produce a ^'text-book 
for students," and the subject-matter has therefore been cut 
down to that which may properly be required of students in 
the time usually allotted to railroad work in a civil-engineenng 
curriculum. On this account no extended discussion has been 
given to the multitudinous forms of various railroad devices 
in the chapters on structures. The aim has been to teach the 
principles and to guide the students into proper methods of 
investigation. 

January, 1903- 



PREFACE TO THIRD EDITION. 



In the present edition Tables IX and X have been entirely 
changed, the tables having~been rewritten so that the values 
are given for single minutes rather than for each ten minutes. 
There has also been added a table of squares, cubes, square 
roots, cube roots, and reciprocals, which are frequently of so 
great service in computations. Advantage has also been taken 
of the opportunity to make numerous typographical and verbal 
changes. 

February, 1905. 



PREFACE TO FOURTH EDITION. 



In this edition a very extensive revision has been made in 
the chapter on Earthwork. Table XXXIII, giving the volume 
of level sections, has been added to the book, with a special 
demonstration of the method of utilizing this table for pre- 



VI PREFACE TO FIFTH EDITION. 

liminary and approximate earthwork calculations. A demon- 
stration, with table, for determining the economics of ties has 
also been added. In accordance with the suggestions of Prof. 
R. B. H. Begg, of Syracuse University, additions have been 
made to Table IV, which facilitate the solution of problems in 
transition curves. Very numerous and sometimes extensive 
alterations and additions, as well as mere verbal and typo- 
graphical changes, have been made in various parts of the 
book. The chapters on Economics have been revised to make 
them conform to more recent estimates of cost of operation. 
July, 1908. 



PREFACE TO FIFTH EDITION. 



The very radical changes made, during recent years, by the 
Interstate Commerce Commission, in the analysis of operating 
expenses of railroads, have required that all of the chapters on 
Economics should be almost entirely rewritten. The writer has 
also endeavored to make all recommendations as to construction 
work to conform with the most recent recommendations on those 
points by the American Railway Engineering Association, who 
granted the Author special permission to make such quotations. 
This gives all such statements the highest possible authority. 
Advantage wr^s taken of the opportunity to correct minor typo- 
graphical errors and blemishes which have been discovered since 
issuing the fourth edition. 

March, 1913. 



TABLE OF CONTENTS, 



CHAPTER I. 

RAILROAD SURVEYS. 

PAGE 

Reconnoissance 1 

1. Character of a reconnoissance survey. 2. Selection of a gen- 
eral route. 3. Valley route. 4. Cross-country route. 5. Moun- 
tain route. 6. Existing maps. 7. Determination of relative 
elevations. Barometrical method. 8. Horizontal measurements, 
bearings, etc. 9. Importance of a good reconnoissance. 

Preliminary surveys 9 

10. Character of a survey. _11. Cross-section method. 12. 
Cross-sectioning. 13. Stadia method. 14. "First" and "sec- 
ond" preliminary survey. 

Location surveys 18 

15. "Paper location." 16. Surveying methods. 17. Form 
of Notes, 

CHAPTER II. 

alignment. 
Simple curves 19 

18. Designation of curves. 19. Length of a subchord. 20. 
Length of a curve. 21. Elements of a curve. 22. Relation be- 
tween T, E, and J. 23. Elements of a 1° curve. 24. Exercises. 
25. Curve location by deflections. 26. Instrumental work. 27. 
Curve location by two transits. 28. Curve location by tangential 
offsets. 29. Curve location by middle ordinates. 30. Curve 
location by offsets from the long chord. 31. Use and value of the 
above methods. 32. Obstacles to location. 33. Modifications of 
location. 34. Limitations in location. 35. Determination of the 
curvature of existing track. 36. Problems. 

Compound curves 38 

< 37. Nature and use. 38. Mutual relations of the parts of a com- 

pound curve having two branches. 39. Modifications of location. 
40. Problems. 

vii 



Vlll TABLE OF CONTENTS, 



PAGE 

Transition curves 43 

41. Superelevation of the outer rail on curves. 42. Practical 
rules for superelevation. 43. Transition from level to inclined 
track. 44. Fundamental principle of transition curves. 45. 
Multiform compound curves. 46. Required length of spiral. 47. 
To find the ordinates of a l°-per-25-feet spiral. 48. To find the 
deflections from any point of the spiral. 49. Connection of spiral 
with circular curve and with tangent. 50. Field-work. 51. To 
replace a simple curve by a curve with spirals. 52. Application 
of transition curves to compound curves. 53. To replace a com- 
pound curve by a curve with spirals. 53a. Use of Table IV. 

Vertical curves 61 

54. Necessity for their use. 55. Required length. 56. Form 
of curve. 57. Numerical example. 

CHAPTER III. 
earthwork. 

Form of excavations and embankments, 65 

58. Usual form of cross-section in cut and fill. 59. Terminal 
pyramids and wedges. 60. Slopes. 61. Compound sections. 
62. Width of roadbed. 63. Form of subgrade. 64. Ditches. 
65. Effect of sodding the slopes, etc. 

Earthwork surveys 73 

66. Relation of actual volume to the numerical result. 67. 
Prismoids. 68. Cross-sectioning 69. Position of slope-stakes. 
70*. Setting slope-stakes by means of "automatic" slope-stake 
rods. 

Computation of volume 79 

71. Prismoidal formula. 72. Averaging end areas. 73. Middle 
areas, 74. Two-level ground. 75. Level sections. 76. Numeri- 
cal example, level sections. 77. Equivalent sections, 78. Equiv- 
alent level sections. 79. Three-level sections. 80. Computation 
of products. 81. Five-level sections. 82, Irregular sections. 
83. Volume of an irregular prismoid. 84. Numerical example; 
irregular sections; volume, with approximate prismoidal correc- 
tion. 85. Magnitude of the probable error of this method. S6. 
Numerical illustration of the accuracy of the approximate rule. 
87. Cross-sectioning irregular sections. 88. Side-hill work. 89. 
Borrow-pits. 90. Correction for curvature. 91, Eccentricity of 
the center of gravity. 92. Center of gravity of side-hill sections. 
93. Example of curvature correction. 94. Accuracy of earthwork 
computations. 95. Approximate computations from profiles. 

Formation of embankments 114 

96. Shrinkage of earthwork. 97. Proper allowance for shrink- 
age. 98. Methods of forming embankments. 

Computation of haul 120 

99. Nature of subject. 100. Mass diagram. 101. Properties 



TABLE OF CONTENTS. Ix 



PAGE 

of the mass curve. 102. Area of the mass curve. 103. Value of 
the mass diagrami. 104. Changing the grade line. 105. Limit of 
free haul. 

Elements op the cost of earthwork 128 

106. Analysis of the total cost into items. 107. Loosening. 
108. Loading. 109. Hauling. 110. Choice of method of haul 
dependent on distance. 111. Spreading. 112. Keeping roadways 
in order. 112a. Trimming cuts to their proper cross-section. 

113. Repairs, wear depreciation, and interest on cost of plant. 

114. Superintendence and incidentals. 115. Contractor's profit 
and contingencies. 116. Limit of profitable haul. 

Blasting 149 

117. Explosives. 118. Drilling. 119. Position and direction 
of drill-holes. 120. Amount of explosive. 121. Tamping. 122. 
Exploding the charge. 123. Cost. 124. Classification of ex- 
cavated material. 125. Specifications for earthwork. 

CHAPTER IV. 

TRESTLES. 

126. Extent of use. 127. Trestles vs. embankments. 128. Two 

principal types I59 

Pile trestles 101 

129. Pile bents. 130. Methods of driving piles. 131. Pile- 
driving formulse. 132. Pile-points and pile-shoes. 133. Details 
of design. 134. Cost of pile trestles. 

Framed trestles 167 

135. Typical design. 136. Joints. 137. Multiple-story con- 
struction. 138. Span. 139. Foundations. 140. Longitudinal 
bracing. 141. Lateral bracing. 142. Abutments. 

Floor systems 173 

143. Stringers. 144. Corbels. 145. Guard-rails. 146 Ties on 
trestles, 147. Superelevation of the outer rail on curves. 148. 
Protection from fire. 149. Timber. 150. Cost of framed timber 
trestles. 

Design of wooden trestles 179 

151. Common practice. 152. Required elements of strength. 
153. Strength of timber. 154. Loading. 155. Factors of safety. 
156. Design of stringers. 157. Design of posts. 158. Design 
of caps and sills. 159. Bracing. 

CHAPTER V. 

tunnels. 
Surveying 189 

160. Surface surveys. 161. Surveying down a shaft. 162. 
Underground surveys. 163. Accuracy of tunnel surveying. 
Design I94 

164. Cross-sections. 165. Grade. 166. Lining. 167. Shafts. 
168. Drains. 



X TABLE OF CONTENTS. 

PAGE 

Construction _ 199 

169. Headings. 170. Enlargement. 171. Distinctive features 
of various methods of construction. 172. Ventilation during con- 
struction. 173. Excavation for the portals. 174. Tunnels vs, 
open cuts. 175. Cost of tunneling. * 

CHAPTER VI. 

CULVERTS AND MINOR BRIDGES. 

176. Definition and object. 177. Elements of the design 207 

Area of the waterway 208 

178. Elements involved. 179. Methods of computation of area. 
180. Empirical formulae. 181. Value of empirical formulae. 182. 
Results based on observation. 183. Degree of accuracy required. 

Pipe culverts 212 

184. Advantages. 185. Construction. 186. Iron-pipe culverts. 
187. Tile-pipe culverts. 

Box CULVERTS 216 

188. Wooden box culverts. 189. Stone box culverts. 190. Old 
rail culverts. 190a. Reinforced concrete culverts. 

Arch culverts 221 

191. Influence of design on flow. 192. Example of arch-cul- 
vert design. 

Minor openings 222 

193. Cattle-guards. 194. Cattle-passes. 195. Standard stringer 
and I-beam bridges. 

CHAPTER VII. 

BALLAST. 

196. Purpose and requirements. 197. Materials. 198. Cross- 
sections. 199. Methods of laying ballast. 200. Cost 227 

CHAPTER VIII. 

TIES AND OTHER FORMS OF RAIL SUPPORT. 

201. Various methods of supporting rails. 202. Economics of 

ties 237 

Wooden ties 238 

203. Choice of wood. 204. Durability. 205. Dimensions. 
206. Spacing. 207. Specifications. 208. Regulations for laying 
and renewing ties. 209. Cost of ties. 

Preservative processes for wooden ties 242 

210. General principle. 211. Vulcanizing. 212. Creosoting. 
213. Burnettizing. 214. Kyanizing. 215. Wellhouse (or zinc- 
tannin) process. 216. Cost of treating. 217. Economics of 
treated ties. 



i 



TABLi: OF CONTENTS. XI 

i»AGE 

Metal, ties 250 

218. Extent of use. 219. Durability. 220. Form and dimen- 
sions of metal cross-ties. 221. Fastenings. 222. Cost. 223. 
Bowls or plates. 224. Longitudinals. 224a. Reinforced concrete 
ties. 

CHAPTER IX. 

RAILS. 

225. Early forms. 226. Present standard forms. 227. "Weight 
for various kinds of traffic. 228. Effect of stiffness on traction. 
229. Length of rails. 230. Expansion of rails. 231. Rules for 
allowing for temperature. 232. Chemical composit'on. 233. 
Proposed standard specifications for steel rails 234, Rf/il wear 
on tangents. 235. Rail wear on curves. 236. Cost cf rails 256 



CHAPTER X. 

rail-fastenings. 

Raiit-joints 270 

237. Theoretical requirements for a perfect joint. 238. Effi- 
ciency of the ordinary angle-bar. 239. Effect of rail-gap at joints. 
240. Supported, suspended, and bridge joints. 241. Failures of 
rail-joints. 242. Standard angle-bars. 243. Later designs of rail- 
joints. 243a. Proposed specifications for steel splice-bars. 

TiE-PLATES ^ 276 

244. Advantages. 245. Elements of the design. 246. Methods, 
of setting. 

Spikes 280 

247. Requirements. 248. Driving. 249. Screws and bolts. 
250. Wooden spikes. 

Track-bolts and nut-locks 284 

251. Essential requirements. 252. Design of track-bolts. 253. 
Design of nut-locks. 

CHAPTER XI. 

switches and crossings. 

Switch construction 289 

254. Essential elements of a switch. 255. Frogs. 256. To find 
the frog number. 257. Stub switches. 258. Point switches. 259. 
Switch-stands. 260. Tie-rods. 261. Guard-rails. 

Mathematical design of switches 297 

262. Design with circular lead rails. 263. Effect of straight frog- 
rails. 264. Effect of straight point-rails. 265. Combined effect 
of straight frog-rails and straight point-rails. 266. Comparison of 
the above methods. 267. Dimensions for a turnout from the 
outer side of a curved track. 268. Dimensions for a turnout from 
the inner side of a curved track. 269. Double turnout from a 



Xll TABLE OF CONTENTS 



PAGE 

straight track. 270. Two turnouts on the same siJe. 271. Con- 
necting curve from a straight track. 272. Connecting curve from 
a curved track to the outside. 273. Connecting curve from a 
curved track to the inside. 274. Crossover between two parallel 
straight tracks. 275. Crossover between two parallel curved 
tracks. 276. Practical rules for switch-laying. 

Crossings 319 

277. Two straight tracks. 278. One straight and one curved 
track. 279. Two curved tracks. 279a. Slips. 

CHAPTER XII. 

/ 

miscellaneous structures and buildings. 

Water stations and water supply. ....... 326 

280. Location. 281. Required qualities of water. 282. Tanks. 
283. Pumping. 284. Track tanks. 285. Stand pipes. 

Buildings 331 

286. Station platforms. 287. Minor stations. 288. Section 
houses. 289. Engine houses. 

Snow structures 336 

290. Snow fences. 291. Snow sheds. 
Turntables •• 338 

CHAPTER XIII. 
yards and terminals. 

293. Value of proper design. 294. Divisions of the subject. . . . 340 

Freight yards 341 

295. General principles. 296. Relation of yard to main tracks. 
297. Minor freight yards. 298. Transfer cranes. 299. Track 
scales. 
Engine yards • 348 

300. General principles. 

Passenger terminals ••• 356 

CHAPTER XIV. 

BLOCK signaling. 

General principles 351 

301. Two fundamental systems. 302. Manual systems. 303. 
Development of the manual system. 304. Permissive blocking. 
305. Automatic systems. 306. Distant signals. 307. Advance 
signals. 

Mechanical details 357 

308. Signals. 309. Wires and pipes. 310. Track circuit for 
automatic signaling. 



TABLE OF CONTENTS. XIU 

CHAPTER XV. 

ROLLING STOCK. 

PAGE 

Wheels and rails 363 

311. Effect of rigidly attaching wheels to their axles. 312. 
Effect of parallel axles. 313. Effect of coning wheels. 314. 
Effect of flanging locomotive driving wheels. 315. Action of a 
locomotive pilot-truck. 

LOCOMOTIVES. 

General structure 37O 

316! Frame. 317. Boiler. 318. Fire box. 319. Coal con- 
sumption. 320. Heating surface. 321. Loss of efficiency of 
steam pressure. 322. Tractive power. 
Running gear, 381 

323. Typ3s of running gear. 324. Equalizing levers. 325. 
Counterbalj^ncing. 326. Mutual relations of the boiler power, 
tractive power and cylinder power for various [types. 327. Life 
of locomotives. 

CARS. 

328. Capacity and size of cars. 329. Stresses to which car- 
frames are subjected. 330. The use of metal. 331. Draft gear. 
332. Gauge of wheels and form of wheel tread 393 

train-brakes. 

333. Introduction. 334. Laws of friction as applied to this 

problem 399 

IHechanism of brakes 403 

335. Hand-brakes. 336. ** Straight" air brakes. 337. Auto- 
matic air brakes. 338. Tests to measure the efficiency of brakes. 
339. Brake shoes. 

CHAPTER XVI. 

train resistance. 

340. Classification of the various forms. 341. Resistances inter- 
nal to the locomotive. 342. Velocity resistances. 343. Wheel 
resistances. 344. Grade resistance. 345. Curve resistance. 346. 
Brake resistances. 347. Inertia resistance. 348. Dynamometer 
tests. 349. Gravity or "drop" tests. 350. Formulae for train 
resistance 409 

CHAPTER XVII. 
cost of railroads. 

351. General considerations. 352. Preliminary financiering. 
; 353. Surveys and engineering e^fpensQs, 354. Land and land 



XIV. TABLE OF CONTENTS 

PAGE 

damages, 355. Clearing and grubbing. 356. Earthwork. 357. 
Bridges, trestles and culverts. 358. Trackwork. 359. Buildings 
and miscellaneous structures. 360. Interest on construction. 
361. Telegraph lines. 362. Detailed estimate of the cost of aline 
of road 42^ 



PART II. 
RAILROAD ECONOMICS. 

CHAPTER XVIII. 

INTRODUCTION. 

363. The magnitude of railroad business. 364. Cost of trans- 
portation. 365. Study of railroad economics — its nature and 
limitations. 366. Outline of the engineer's duties. 367, Justi- 
fication of such methods of computation 436 

CHAPTER XIX. 

THE PROMOTION OF RAILROAD PROJECTS. 

368. Method of formation of railroad corporations. 369. The 
two classes of financial interests, the security and profits of each. 

370. The small margin between profit and loss to the projectors. 

371. Extent to which a railroad is a monopoly. 372. Profit 
resulting from an increase in business done; loss resulting from a 
decrease. 373. Estimation of probable volume of traffic, and of 
probable growth. 374. Probable number of trains per day. In- 
crease with growth of traffic. 375. Effect on traffic of an increase 
in facilities. 376. Loss caused by inconvenient terminals and 
by stations far removed from business centres. 377. General 
principles which should govern the expenditure of money for 
railroad purposes 44 J 



CHAPTER XX. 

OPERATING EXPENSES. 

378. Distribution of gross revenue. 379. Fivefold distribution 
of operating expenses. 380. Operating expenses per train mile. 
381. Reasons for uniformity in expenses per train mile. 382. 
Detailed classification of expenses with ratios to the total expense. 
383. Amounts and percentages of the various items 454 



TABLE OF CONTENTS. XV 

PAGE 

Maintenance of way and structures 463 

384. Track material. 385. Roadway and track, 386. Main- 
tenance of track structures. 387. Other items. 

Maintenance of equipment 466 

388. Superintendence. 389. Repairs, renewals, and deprecia- 
tion of steam locomotives. 390. Repairs, renewals and depreciation 
of electric locomotives. 391. Repairs, renewals and depreciation of 
passenger cars, and of freight cars. 392. Electric equipment; 
marine equipment; shop machinery and tools; stationery and 
printing; other expenses. 393. Traffic. 

Transportation 467 

394. Superintendence. 395. Yard engine expenses. 396. Road 
enginemen. 397. Fuel for road locomotives. 398. Water, lubri- 
cants, and other supplies for road locomotives. 399. Road train- 
men. 400. Train supplies and expenses. 401. Clearing wrecks, 
loss, damage and injuries to persons and property. 402. Operating 
joint tracks and facilities. 

CHAPTER XXI. 

distance. 

403. Relation of distance to rates and expenses. 404. The 
conditions other than distance that affect the cost; reasons why 
rates are usually based on distance 474 

Effect of distance on operating expenses 475 

405. Effect of slight changes in distance on maintenance of way. 
406. Effect on maintenance of equipment. 407. Effect on con- 
ducting transportation. 408. Estimate of total effect on expenses 
of small changes in distance (measured in feet) ; estimate for dis- 
tances measured in miles. 

Effect of distance on receipts 486 

409. Classification of traffic. 410. Method of division of through 
rates between the roads run over. 411. Effect of a change in the 
length of the home road on its receipts from through competitive 
traffic. 412. The most advantageous conditions for roads forming 
part of a through competitive route. 413. Effect of the variations 
in the length of haul and the classes of the business actually done. 
414. General conclusions regarding a change in distance. 415. 
Justification of decreasing distance to save time. 416. Effect of 
change of distance on the business done. 



CHAPTER XXII. 

curvature. 

417. General objections to curvature. 418. Financial value of 
the danger of accident due to curvature. 419. Effect of curvature 
on travel. 420. Effect on operation of trains 492 



XVI TABLE OF CONTENTS. 



PAGE 

Effect of Curvature on operating expenses 496 

421. Relation of radius of curvature and of degrees of central 
angle to operating expenses. 422. Effect of curvature on mainte- 
nance of way. 423. Effect of curvature on maintenance of equip- 
ment. 424. Effect of curvature on conducting transportation. 
,425. Estimate of total effect per degree of central angle. 426. 
Reliability and value of the above estimate. 

Compensation for curvature. 504 

427. Reasons for compensation. 428. The proper rate of com- 
pensation. 429. The limitations of maximum curvature. 



CHAPTER XXIII. 

GRADE. 

430. Two distinct effects of grade. 431. Application to the 
movement of trains of the laws of accelerated motion. 432. Con- 
struction of a virtual profile. 433. Use value and possible misuse. 
434. Undulatory grades; advantages, disadvantages, and safe 
limits 509 

Minor grades 516 

435. Basis of cost of minor grades. 436. Classification of minor 
grades. 437. Effect on operating expenses. 438. Estimate of 
the cost of one foot of change of elevation. 439. Operating value 
of the filling of a sag in a grade. 

Ruling grades . ; 523 

440. Definition. 441, Choice of ruling grades. 442. Maximum 
train load on any grade. 443. Proportion of traffic affected by 
the ruling grade. 444. Financial value of increasing the train 
load. 445. Operating value of a reduction in the rate of the ruling 
grade. 

Pusher grades 532 

446. General principles underlying the use of pusher engines. 
447. Balance of grades for pusher service. 448. Operation of 
pusher engines. 449. Length of a pusher grade. 450. Cost of 
pusher engine service. 451. Numerical comparison of pusher and 
through grades. 

Balance of grades for unequal traffic 542 

452. Nature of the subject. 453. Computation of the theoreti- 
cal balance. 454. Computation of relative traffic. 

CHAPTER XXIV. 

the improvement of old lines. 

455. Classification of improvements. 456. Advantages of re- 
locations. 457. Disadvantages of re-locations 546 

Reduction of virtual grade 549 

458. Obtaining data for computations. 459, Use of the data 
obtained. 460. Reducing the starting grade at stations. 



TABLE OF CONTENTS. XVll 

PAGE 

Appendix. The adjustments of instruments 554 

Tables. 

I. Radii of curves 564 

II. Tangents, external distances, and long chords for a 1° curve. 568 

III. Switch leads and distances 571 

IV. Transition curves 573 

V. Logarithms of numbers 582 

VI. Logarithmic sines and tangents of small angles 602 

VII. Logarithmic sines, cosines, tangents, and cotangents 605 

VIII. Logarithmic versed sines and external secants 650 

IX. Natural sines, cosines, tangents, and cotangents 695 

X. Natural versed sines and external secants 718 

XI. Reduction of barometer reading to 32° F 741 

XII. Barometric elevations 742 

XIII. Coefficients for corrections for temperature and humidity. . 742 

XIV. Capacity of cylindrical water-tanks in United States standard 

gallons of 231 cubic inches 329 

XV. Number of cross ties per mile 430 

XVI. Tons per mile (with cost) of rails of various weights 431 

XVII. Splice bars and bolts per mile of track 432 

XVIII. Railroad spikes 433 

XIX. Track bolts 433 

XX. Classification of operating expenses of all railroads 470-473 

XXI. Effect on operating expenses of changes in distance 485 

XXII. Effect on operating expenses of changes in curvature 501 

XXIII. Velocity head of trains .__. 512 

XXIV. Effect .on operating expenses of changes in grade 520 

XXV. Tractive power of locomotives 525 

XXVI. Total train resistance per ton on various grades 527 

XXVII. Cost of an additional train to handle a given traffic 532 

XXVIII. Balanced grades for one, two and three engines 536 

XXIX. Cost for each mile of pusher engine service 540 

XXX. Useful trigonometrical formulae 743 

XXXI. Useful formulae and cons-tants 745 

XXXII. Squares, cubes, square roots, cube roots and reciprocals 746 

XXXIII. Cubic yards per 100 feet of level sections 763 

XXXIVc Annual charge against a tie, based on the original cost and as- 
sumed life of the tie 766 

Index 767 



I 



RATLEOAD CONSTRUCTION. 



CHAPTER I. 

RAILROAD SURVEYS 

The proper conduct of railroad surveys presupposes an 
adequate knowledge of almost the whole subject of railioad 
engineering, and particularly of some of the complicated ques- 
tions of Railroad Economics, which are not generally studied 
except at the latter part of a course in railroad engineering, if 
at all. This chapter will therefore be chiefly devoted to methods 
of instrumental work, and the problem of choosing a general 
route will be considered only as it is influenced by the topog- 
raphy or by the application of those elementary principles of 
Railroad Economics which are self-evident or which may be 
accepted by the student until he has had an opportunity of 
studying those principles in detail 

RECONNOISSANCE SURVEYS. 

1. Character of a reconnoissance survey. A reconnoissance 
survey is a very hasty examination of a belt of country to de- 
termine which of all possible or suggested routes is the most 
promising and best worthy of a more detailed survey. It is 
essentially very rough and rapid. It aim^s to discover those 
salient features which instantly stamp one route as distinctly 
superior to another and so narrow the choice to routes which 
are so nearly equal in value that a more detailed survey is nec- 
essary to decide between them. 

2. Selection of a general route. The general question of 
running a railroad between two towns is usually a financial rather 



2 RAILROAD CONSTRUCTION. § 3. 

than an engineering question. Financial considerations usually 
determine that a road must pass through certain more or less 
important towns between its termini. When a railroad runs 
through a thickly settled and very flat country, where, from a 
topographical standpoint, the road may be run by any desired 
route, the "right-of-way agenf sometimes has a greater influ- 
ence in locating the road than the engineer. But such modifi- 
cations of alignment, on account of business considerations, are 
foreign to the engineer's side of the subject, and it will be here- 
after assumed that topography alone determines the location of 
the line. The consideration of those larger questions combin- 
ing finance and engineering (such as passing by a town on ac- 
count of the necessary introduction of heavy grades in order to 
reach it) will be taken up in Chap. XIX, et seq. 

3. Valley route. This is perhaps the simplest problem. If 
the two towns to be connected lie in the same valley, it is fre- 
quently only necessary to run a line which shall have a nearly 
uniform grade. The reconnoissance problem consists largely in 
determining the difference of elevation of the two termini of 
this division and the approximate horizontal distance so that the 
proper grade may be chosen. If there is a large river running 
through the valley, the road will probably remain on one side 
or the other throughout the whole distance, and both banks 
should be examined by the reconnoissance party to determine 
which is preferable. If the river may be easily bridged, both 
banks may be alternately used, especially when better alignment 
is thereby secured. A river valley has usually a steeper slope 
in the upper part than in the lower part. A uniform grade 
throughout the valley will therefore require that the road climbs 
up the side slopes in the lower part of the valley. In case the 
'^ruling grade'' * for the whole road is as great as or greater 
than the steepest natural valley slope, more freedom may be 
used in adopting that alignment which has the least cost — 
regardless of grade. The natural slope of large rivers is almost 
invariably so low that grade has no influence in determining the 
choice of location. When bridging is necessary, the river 
banks should be examined for suitable locations for abutments 



* The ruling grade may here be loosely defined as the maximum grade 
which is permissible. This definition is not strictly true, as may be seen later 
when studying Railroad Economics, but it may here serve the purpose. 



§ 4. RAILROAD SURVEYS. 3 

and piers. If the soil is soft and treacherous, much difficulty- 
may be experienced and the choice of route may be largely 
determined by the difficulty of bridging the river except at 
certain favorable places. 

4. Cross-country route. A cross-country route always has one 
or more summits to be crossed. The problem becomes more 
complex on account of the greater number of possible solutions 
and the difficulty of properly weighing the advantages and dis- 
advantages of each. The general aim should be to choose the 
lowest summits and the highest stream crossings, provided that 
by so doing the grades between these determining points shall 
be as low as possible and shall not be greater than the ruling 
grade of the road. Nearly all railroads combine cross-country 
and valley routes to some extent. Usually the steepest natural 
slopes are to be found on the cross-country routes, and also the 
greatest difficulty in securing a low through grade. An approx- 
imate determination of the ruling grade is usuall}^ made during 
the reconnoissance. If the ruling grade has been pre\dously 
decided on by other considerations, the leading feature of the 
reconnoissance survey mil be the determination of a general 
route along which it will be possible to survey a line whose 
maximum grade shall not exceed the ruling grade. 

5. Mountain route. The streams of a mountainous region 
frequently have a slope exceeding the desired ruling grade. In 
such cases there is no possibility of securing the desired grade 
by following the streams. The penetration of such a region 
may only be accomplished by ^^development" — accompanied 
perhaps by tiumeling. ^'Development" consists in deliber- 
ately increasing the length of the road between two extremes 
of elevation so that the rate of grade shall be as low as desired. 
The usual method of accomplishing this is to take advantage of 
some convenient formation of the ground to introduce some 
lateral deviation. The methods may be somewhat classified as 
follows : 

(a) Running the fine up a convenient lateral valle}^, turning 
a sharp curve and working back up the opposite slope. As 
sho^ii in Fig. 1, the considerable rise between A and B was 
surmounted by starting off in a very different direction from 
the general direction of the road; then, w^hen about one-half of 
the desired rise had been obtained, the line crossed the valley 
and continued the climb along the opposite slope, (b) Switch- 



RAILROAD CONSTRUCTION. 



§5. 



hack. On the steep side-hill BCD (Fig. 1) a very considerable 
gain in elevation was accomplished by the switchback CD, 
The gain in elevation from B to D is very great. On the other 
hand, the speed must always be slow; there are two complete 
stoppages of the train for each run; all trains must run back- 
ward from C to D. (c) Bridge spiral. When a valley is so 
narrow at some point that a bridge or viaduct of leasonable 
length can span the valley at a considerable elevation above the 




Fig. 1. 

bottom of the valley, a bridge spiral may be desirable. In Fig. 2 
the line ascends the stream valley past A, crosses the stream At 
Bj works back to the narrow place at C, and there crosses itself, 
having gained perhaps 100 feet in elevation, (d) Tunne\ 
spiral (Fig. 3). This is the reverse of the previous plan. It 
implies a thin steep ridge, so thin at some place that a tunned 
through it will not be excessively long. Switchbacks and 
spirals are sometimes necessary in mountainous countries, but 
they should not be considered as normal types of construction. 
A region must be very difficult if these devices cannot be avoided. 
On Plate I are shown three separate ways (as actually con- 
structed) of running a railroad between two points a little over 
three miles apart and having a difference of elevation of nearly 



§5. 



RAILROAD SURVEYS. 



1100 feet. At A the Central R. R. of New Jersey runs under 
the Lehigh Valley R. R. and soon turns off to the northeast for 
about six miles, then doubles back, reaching D, a fall of about 
1050 feet with a track distance of about 12.7 miles. The 
L. V. R. R. at A nms to the westw^ard for six to seven miles, 





Fig. 2. 



Fig. 3. 



then turns back until the roads are again close together at D. 
The track distance is about 14 miles and the drop a little greater, 
since at A the L. V. R. R. crosses over the other, while at D they 
are at practically the same level. From B to C the distance is 
over eleven miles. From A directly down to D the C. R. R. of 
N. J. runs a ^'gravity'' road, used exclusively for freight, on 
w^hich cars alone are hauled by cable. The main-line routes 
are remarkable examples of sheer ''development.'' Even as 
constructed the L. V. R. R. has a grade of about 95 feet per 
mile, and this grade has proved so excessive for freight work 
that the company has constructed a cut-off (not shown on the 
map) which leaves the main line at A, nearly parallels the 
C. R. R. to C, and then running in a northeasterly direction 
again joins the main line beyond Wilkesbarre. The grade is 
thereby cut down to 65* feet per mile. 

Rack railways and cable roads, although types of mountain 
railroad construction, will not be here considered. 



6 



RAILROAD CONSTRtrCTION. 



§6. 



6. Existing maps. The maps of the U. S. Geological Survey 
are exceedingly valuable as far as they have been completed. 
So far as topographical considerations are concerned, they 
almost dispense with the necessity for the reconnoissance and 
''first preliminary" surveys. Some of the State Survey maps 
will give practically the same information. County and town- 
ship maps can often be used for considerable information as to the 
relative horizontal position of governing points, and even some 
approximate data regarding elevations may be obtained by a 
study of the streams. Of course such information will not dis- 
pense with surveys, but will assist in so planning them as to 
obtain the best information with the least work. When the 
relative horizontal positions of points are reliably indicated on o. 
map, the reconnoissance may be reduced to the determination 
of the relative elevations of the governing points of the route. 

7. Determination of relative elevations. A recent description 
of European methods includes spirit-leveling in the reconnois- 
sance work. This may be due to the fact that, as indicated 
above, previous topographical surveys have rendered unnecessary 
the "exploratory" survey which is required in a new country, 
and that their reconnoissance really corresponds more nearly to 
our preliminary. 

The perfection to which barometrical methods have been 
brought has rendered it possible to determine differences of 
elevation with sufficient accuracy for reconnoissance purposes 
by the combined use of a mercurial and an aneroid barometer. 
The mercurial barometer should be kept at "headquarters," and 
readings should be taken on it at such frequent intervals that 
any fluctuation is noted, and throughout the period that observa- 
tions with the aneroid are taken in the field. At each observa- 
tion there should also be recorded the time, the reading of the 
attached thermometer, and the temperature of the external 
air. For uniformity, the mercurial readings should then be 
"reduced to 32° F." The form of notes for the mercurial 
barometer readings should be as follows : 



Time. 


Merc. 
Barom. 


Attached 
Therm. 


Reduction 
to 32° F. 


External 
Therm. 


Corrected 
reading. 


7:00 A.M. 
:15 
:30 
:45 


29.872 
.866 
.858 
.850 


72° 
73.5 
75 
76 


— .11/ 
.121 
.125 
.127 


73° 
75 

76 

77 


29.755 
.745 
.733 
.723 



§ 7. EAILROAD SURVEYS. 7 

The corrections in column 4 are derived from Table XI by 
interpolatioQ. 

Before starting out, a reading of the aneroid should be taken 
at headquarters coincident with a reading of the mercurial. 
The difference is one value of the correction to the aneroid. 
As soon as the aneroid is brought back another comparison of 
readings should be made. Even though there has been con- 
siderable rise or fall of pressure in the interval, the difference 
in readings (the correction) should be substantially the same 
provided the aneroid is a good instrument. If the difference 
of elevation is excessive (as when climbing a high mountain) 
even the best aneroid mil "lag'' and not recover its normal 
reading for several hours, but this does not apply to such dif- 
ferences of elevation as are met with in railroad w^ork. The 
best aneroids read directly to -^ of an inch of mercury and 
may be estimated to yiroir ^^ ^^ inch — ^which corresponds 
to about 0.9 foot difference of elevation. In the field there 
should be read, at each point whose elevation is desired, the 
aneroid, the time, and the temperature. These readings, cor- 
rected by the mean value of the correction between the aneroid 
and the mercurial, should then be combined with the reading 
of the mercurial (interpolated if necessary) for the times of 
the aneroid observations and the difference of elevation ob- 
tained. The field notes for the aneroid should be takeji as 
shown in. the first- four colunms of the tabular form. The " cor- 
rected aneroid" readings of column 5 are found by correcting 
the readings of column 3 by the mean difference betw^een the 
mercurial and aneroid w^hen compared at morning and night. 
Column 6 is a copy of the "corrected readings" from the office 
notes, interpolated when necessary for the proper time. Column 
7 is similarly obtained. Col. 8 is obtained from cols. 4 and 5, 
and col. 9 from cols. 6 and 7, with the aid, of Table XII. The 
correction for temperature (col. 11), w^hich is generally small 
unless the difference of elevation is large, is obtained with the 
aid of Table XIII. The elevations in Table XII are elevations 
above an assumed datum plane, where under the given atmos- 
pheric conditions the mercurial reading would be 30". Of 
comrse the position of this assumed plane changes wdth varying 
atmospheric conditions and so the elevations are to be con- 
sidered as relative and their difference taken. [See the author's 
"Problems in the Use and Adjustment of Engineering In- 



8 



EAILROAD CONSTRUCTION. 



§8. 



(Left-hand page of Notes.) 



Time. 


Place. 


Aneroid. 


Therm. 


Corr. 
Aner. 


Corr. 
Merc. 


7 00 


Office 
JO 
saddle-back 
river cross. 


29 . 628 
29.662 
29.374 
29.548 


73° 

72° 
63° 
70° 




29 755 


7:10 
7:30 
7:50 


29.789 
29.501 
29.675 


29.748 
29.733 
29.720 



struments/^ Prob. 22.] Important points should be observed 
more than once if possible. Such duplicate observations will be 
found to give surprisingly concordant results even when a 
general fluctuation of atmospheric pressure so modifies the 
tabulated readings that an agreement is not at first apparent. 
Variations of pressure produced by high winds, thunder-storms, 
etc., will generally vitiate possible accuracy by this method. 
By "headquarters" is meant any place whose elevation above 
any given datum is known and where the mercurial may be 
placed and observed while observations within a range of several 
miles are made with the aneroid. If necessary, the elevation of 
a new headquarters may be determined by the above method, 
but there should be if possible several independent observations 
whose accordance will give a fair idea of their accuracy. 

The above method should be neither slighted nor used for 
more than it is worth. When properly used, the errors are 
compensating rather than cumulative. When Used, for example, 
to determine that a pass B is 260 feet higher than a determined 
bridge crossing at A which is six miles distant, and that another 
pass C is 310 feet higher than A and is ten miles distant, the 
figures, even with all necessary allowances for inaccuracy, .will 
give an engineer a good idea as to the choice of route especially 
as affected by ruling grade. There is no comparison between 
the time and labor involved in obtaining the above information 
by barometric and by spirit-leveling methods, and for recon" 
noissance purposes the added accuracy of the spirit-leveling 
method is hardly worth its cost. 

8. Horizontal measurements, bearings, etc. When there is 
no map which may be depended on, or when only a skeleton 
map is obtainable, a rapid survey, sufficiently accurate for the 
purpose, may be made by using a pocket compass for bearings 
and a telemeter, odometer, or pedometer for . distances. The 
telemeter [stadia] is more accurate, but it requires a definite cleay 



9. 



RAILROAD SURVEYS. 



(Right-hand page of Notes.) 



Temp, at 
headqu. 


Approx. 
field read. 


Approx. 
headq. read. 


Diff. 


Corr. for 
temp. 


Diff. 
elev. 


75« 

76 

77 


192 
457 
297 


230 
244 
256 


- 38 
+ 213 
+ 41 


+ ( + 10) 
+ (+ 2) 


— 40 
+ 223 
+ 43 



sight from station to station, which may be difficult through a 
wooded country. The odometer, which records the revolutions 
of a wheel of known circumference, may be used even in rough 
and wooded country, and the results may be depended on to a 
small percentage. The pedometer (or pace-measurer) depends 
for its accuracy on the actual movement of the mechanism for 
each pace and on the uniformity of the pacing. Its results are 
necessarily rough and approximate, but it may be used to fill 
in some intermediate points in a large skeleton map. A hand- 
level is also useful in determining the relative elevation of various 
topographical features which may have some bearing on the 
proper location of the road. 

9. Importance of a good reconnoissance. The foregoing in- 
struments and methods should be considered only as aids in 
exercising an educated common sense, without which a proper 
location cannot be made. The reconnoissance survey should 
command the best talent and the greatest experience available. 
If the general route is properly chosen, a comparatively low 
order of engineering skill can fill in a location which will prove 
a paying railroad property ; but if the general route is so chosen 
that the ruling grades are high and the business obtained is small 
and subject to competition, no amoimt of perfection in detailed 
ahgnment or roadbed construction can make the road a profitable 
investment. 



PRELIMINARY SUR^^YS. 

10. Character of survey. A preliminary railroad survey is 
properly a topographical survey of a belt of country which has 
been selected during the reconnoissance and within which it is 
estimated that the located line will He. The width of this belt 
will depend on the character of the country. When a railroad 
is to follow a river having very steep banks the choice of loca- 
tion is sometimes limited at places to a very few feet of width 



10 



KAILROAD CONSTRUCTION. 



§ 11 



and the belt to be surveyed may be correspondingly narrowed. 
In very flat country the desired width may be only limited by the 
ability to survey points with sufficient accuracy at a considerable 
distance from what may be called the '^backbone line" of the 
survey. 

II. Cross-section method. This is the only feasible method 
in a wooded country, and is employed by many for all kinds 
of country. The backbone line is surveyed either by observ- 
ing magnetic bearings with a compass or by carrying forward 




Fig. 4. 



absolute azimuths with a transit. The compass method nas 
the disadvantages of limited accuracy and the possibiHty of 
considerable local error owing to local attraction. On the other 
hand there are the advantages of greater simplicity, no necessity 



§ 12. RAILROAD SURVEYS. 11 

for a back rodman, and the fact that the errors are purely- 
local and not cumulative, and may be so limited, with care, that 
they will cause no vital error in the subsequent location survey. 
The transit method is essentially more accurate, but is liable 
to be more laborious and troublesome. If a large tree is en- 
countered, either it must be cut down or a troublesome opera- 
tion of offsetting must be used. If the compass is employed 
under these circumstances, it need only be set up on the far side 
of the tree and the former bearing produced. An error in 
reading a transit azimuth will be carried on throughout the 
survey. An error of only five minutes of arc will cause an off- 
set of nearly eight feet in a mile. Large azimuth errors may, 
however, be avoided by immediately checking each new azimuth 
with a needle reading. It is advisable to obtain true azimuth 
at the beginning of the survey by an observation on the sun * or 
Polaris, and to check the azimuths every few miles by azimuth 
observations. Distances along the backbone line should be 
measured with a chain or steel tape and stakes set every 100 
.feet. When a course ends at a substation, as is usually the case, 
the remaining portion of the 100 feet should be measured along 
the next course. The level party should immediately obtain the 
elevations (to the nearest tenth of a foot) of all stations, and also 
of the lowest points of all streams crossed and even of dry gullies 
which would require culverts. 

12. Cross-sectioning. It is usually desirable to obtain con- 
tours at five-foot intervals This may readily be done by the 
use of a Locke level (which should be held on top of a simple 
five-foot stick), a tape, and a rod ten feet in length graduated 
to feet and tenths. The method of use may perhaps be best 
explained by an example. Let Fig. 5 represent a section per- 
pendicular to the survey line — such a section as would be made 
by the dotted lines in Fig. 4. C represents the station point. 
Its elevation as determined by the level is, say, 158.3 above 
datum. When the Locke level on its five-foot rod is placed at 
Cf the level has an elevation of 163.3. Therefore when a point 
is found (as at a) where the level will read 3.3 on the rod, that 
point has an elevation of 160.0 and its distance from the center 
gives the position of the 160-foot contour. Leaving the long 
rod at that point (a), carry the level to some point (b) such that 
the level will sight at the top of the rod. b is then on the 165- 

* For detailed methods of such determinations, see the author's '* Problems 
in the Use and Adjustment of Engineering Instruments," Problems 35 and 36. 



12 



KAILROAD CONSTRUCTION. 



§12, 



foot contour, and the horizontal distance ah added to the hori- 
zontal distance ac gives the position of that contour from the 
center. The contours on the lower side are found similarly. 
The first rod reading will be 8.3, giving the 155-foot contour. 




Fig. 5. 

Plot the results in a note-book which is ruled in quarter-inch 
squares, using a scale of 100 feet per inch in both directions. 
Plot the work up the page; then when looking ahead along the 
line, the work is properly oriented. When a contour crosses 




Fig. 6. 

the survey line, the place of crossing may be similarly deter- 
mined. If the ground flattens out so that five-foot contours are 
very far apart, the absolute elevations of points at even fifty- , 



§ 13. RAILROAD SURVEYS. 13 

foot distances from the center should be determined. The 
method is exceedmgly rapid. Whatever error or inaccuracy 
occurs is confined in its effect to the one station where it occurs. 
The work being thus plotted in the field, unusually irregular 
topography may l)e plotted Avith greater certainty and no great 
error can occur without detection. It w^ould even be possible 
by this method to detect a gross error that might have been 
made by the level party 

13. Stadia method. This method is best adapted to fairly 
open country where a ^'shot" to any desired point may be 
taken without clearing. The hackhone survey line is the same 
as in the previous method except that each course is limited to 
the practicable length of a stadia sight. The distance between 
stations should be checked by foresight and backsight — also the 
vertical angle. Azimuths should be checked by the needle. 
Considering the vital importance of leveling on a railroad surA'ey 
it might be considered desirable to run a line of levels over the 
stadia stations m order that the leveling may be as precise as 
possible; but when it is considered that a preliminary survey is 
a somew^hat hasty survey of a route that may be abandoned, and 
that the errors of leveling b}^ the stadia method (which are con- 
pensating) may be so minimized that no proposed route would 
be abandoned on account of such small error, and that the effect 
of such an error may be easily neutralized by a slight change in 
the location, it may be seen that excessive care in the leveling 
of the preliminary survey is hardly justifiable. 

Since the students taking this w^ork are assumed to be familiar 
wnth the methods of stadia topographical surve3'S, this part of 
the subject will not be further elaborated. 

14. " First " and " Sscond '' preliminary surveys. Some engi- 
neers advocate tw^o preliminary surveys. When this is done, 
the first IS a very rapid survey, made perhaps with a compass, 
ard is only a better grade of reconnoissance. Its aim is to 
rapidl v' develop the facts which W'ill decide for or against any 
proposed route, so that if a route is found to be unfavorable 
another more or less modified route may be adopted Avithout 
having wasted considerable time in the survey of useless details. 
By this time the student should have grasped the fundamental 
idea that both the reconnoissance and preliminary surveys are 
not surveys of lines but of areas; that their aim is to survey 
only those topographical features which w^ould have a deter- 



14 RAILROAD CONSTRUCTION. § 15. 

mining influence on any railroad line which might be constructed 
through that particular territory, and that the value of a locating 
engineer is largely measured by his ability to recognize those 
determining influences with the least amount of work from his 
surveying corps. Frequently too little time is spent on the 
comparative study of preliminary lines. A line will be hastily 
decided on after very little study; it will then be surveyed with 
minute detail and estimates carefully worked up, and the claims 
of any other suggested route will then be handicapped, if not 
disregarded, owing to an unwillingness to discredit and throw 
away a large amount of detailed surveying. The cost of two or 
three extra preliminary surveys {at critical sections and not over 
the whole line) is utterly insignificant compared with the prob- 
able improvement in the ''operating value'' of a line located 
after such a comparative study of preliminary lines. 

LOCATION SURVEYS. 

15. "Paper location." When the preliminary survey has 
been plotted to a scale of 200 feet per inch and the contours 
drawn in, a study may be made for the location survey. Disre- 
garding for the present the effect on location of transition curves, 
the alignment may be said to consist of straight lines (or "tan- 
gents'') and circular curves. The ''paper location" therefore 
consists in plotting on the preliminary map a succession of 
straight lines which are tangent to the circular curves connect- 
ing them. The determining points should first be considered. 
Such points are the termini of the road, the lowest practicable 
point over a summit, a river-crossing, etc. So far as is possi- 
ble, having due regard to other considerations, the road should 
be a "surface" road, i.e., the cut and fill should be made as 
small as possible. The maximum permissible grade must also 
have been determined and duly considered. The method of 
location differs radically according as the lines joining the deter- 
mining points have a very low grade or have a grade that ap- 
proaches the maximum permissible. With very low natural 
grades it is only necessary to strike a proper balance between 
the requirements for easy alignment and the avoidance of exces-3 
sive earthwork. When the grade betw^een two determined 
points approaches the maximum, a study of the location may b^ 
begun by finding a strictly surface line which will connect tho 



§ 16. RAILROAD SURVEYS. 15 

points with a line at the given grade. For example, suppose 
the required grade is 1.6% and that the contours are drawn at 
5-foot intervals It will require 312 feet of 1.6% grade to rise 
5 feet. Set a pair of dividers at 312 feet and step off this in- 
terval on successive contours. This line will in general be very 
irregular, but in an easy country it may lie fairly close to the 
proper location line, and even in difficult country such a surface 
line will assist greatly in selecting a suitable location. When the 
larger part of the line will evidently consist of tangents, the tan- 
gents should be first located and should then be connected by 
suitable curves. When the curves predominate, as they gener- 
ally will in mountainous country, and particularly when the line 
is purposely lengthened in order to reduce the grade, the curves 
should be plotted first and the tangents may then be drawn 
connecting them. Considering the ease with which such lines 
may be drawTi on the preliminary map, it is frequently advisable, 
after making such a paper location, to begin all over, draw a 
new line over some specially difficult section and compare re- 
sults. Profiles of such lines may be readily drawn by noting their 
intersection with each contour crossed. Drawing on each profile 
the required grade line will furnish an approximate idea of the 
comparative amount of earthwork required. After deciding on 
the paper location, the length of^ each tangent, the central angle 
(see § 21), and the radius of each curve should be measured as 
accurately as possible. Since a slight error made in such meas- 
urements, taken from a map with a scale of 200 feet per inch, 
would by accumulation cause serious discrepancies between the 
plotted location and the location as afterward surveyed in the 
field, frequent tie lines and angles should be determined between 
the plotted location line and the preliminary line, and the loca-.' 
tion should be altered, as may prove necessary, by changing the 
length of a tangent or changing the central angle or radius of a 
curve, so that the agreement of the check-points will be suffi- 
ciently close. The errors of an inaccurate preliminary survey 
may thus be easily neutralized (see § 33). When the pre- 
liminary line has been properly run, its '' backbone'' line will 
lie very near the location line and will probably cross it at fre- 
quent intervals, thus rendering it easy to obtain short and nu- 
merous tie lines. 

1 6. Surveying methods. A transit should be used for align- 
ment, and only precise work is allowable. The transit stations 



16 



RAILROAD CONSTRUCTION. 



§16. 



should be centered with tacks and should be tied to witness- 
stakes, which should be located outside of the range of the earth- 
work, so that they will neither be dug up nor covered up. All 
original property lines lying within the limits of the right of way 
should be surveyed with reference to the location hne, so that 
the right-of-way agent may have a propei basis for settlement. 
When the property lines do not extend far outside of the re- 
quired right of way they are frequently surveyed completely. 

The leveler usually reads the target to the nearest thousandth 
of a foot on turning-points and bench-marks, but reads to the 
nearest tenth of a foot for the elevation of the ground at stations. 
Considering that y-oVu of a foot has an angular value of about 



FORM OF NOTES. 



[Left-hand page.] 



Sta. 


Align- 
ment. 


Vernier. 


Tan2:ential 
Deflection. 


Calculated 
Bearing. 


Needle. 


54 












53 












-^72.2 


P.T. 


9<^11' 


18° 22' 


N 54° 48' E 


N 62° 15' E 


52 


1 1 


7 57 








51 




6 15 








50 


> G 

^5 


4 33 








49 


^00 


2 51 








48 


1 1 


1 09 








0+32 


P.O. 


0° 








47 












46 








N 36° 26' E 


N 44° 0' E 



§16. 



RAILROAD SURVEYS. 



17 



one second at a distance of 200 feet, and that one division of a level- 
bubble is usually about 30 seconds, it ma}^ be seen that it is a 
useless refinement to read to thousandths unless corresponding 
care is taken in the use of the level. The leveler should also 
locate his bench-marks outside of the range of earthwork. A 
knob of rock protruding from the ground affords an excellent 
mark. A large nail, driven in the roots of a tree, which is not 
to be disturbed, is also a good mark. These marks should be 
clearly described in the note-book. The leveler should obtain 
the elevation of the ground at all station-points; also at all 
sudden breaks in the profile line, determining also the distance 
of these breaks from the previous even station. This will in- 



[Right-hand page.] 





wm. brown 



JOHN JONES 



18 RAILROAD CONSTRUCTION, § 17. 

elude the position and elevation of all streams, and even dry 
gullies, which are crossed 

Measurements should preferably be made with a steel tape, 
care being taken on steep ground to insure horizontal measure- 
ments. Stakes are set each 100 feet, and also at the beginning 
and end of all curves. Transit-points (sometimes called '' plugs" 
or ''hubs") should be driven flush with the ground, and a 
^'witness-stake," having the "number " of the station, should 
be set three feet to the right. For example, the witness-stake 
might have on one side "137 + 69.92," and on the other side 
"PC4°R," which would signify that the transit hub is 69.92 
feet beyond station 137, or 13769.92 feet from the beginning of 
the line, and also that it is the "point of curve" of a "4° curve" 
which turns to the right. 

Alignment. The alignment is evidently a part of the loca- 
tion survey, but, on account of the magnitude and importance 
of the subject, it will be treated in a separate chapter. 

17. Form of Notes. Although the Form of Notes cannot be 
thoroughly understood until after curves are studied, it is here 
introduced as being the most convenient place. The right-hand 
page should have a sketch showing all roads, streams, and 
property lines crossed with the bearings of those lines. This 
should be drawn to a scale of 100 feet per inch — the quarter- 
inch squares which are usually ruled in note-books giving con- 
venient 25-foot spaces. This sketch will always be more or less 
distorted on curves, since the center line is always shown as 
straight regardless of curves. The station points ("Sta." in 
first column, left-hand page) should be placed opposite to their 
sketched positions, which means that even stations will be 
recorded on every fourth line. This allows three intermediate 
lines for substations, which is ordinarily more than sufficient. 
The notes should read up the page, so that the sketch will be 
properly oriented when looking ahead along the line The 
other columns on the left-hand page will be self-explanatory 
when the subject of curves is understood. If the "calculated J 
bearings" are based on azimuthal observations, their agreement ' 
(or constant difference) with the needle readings will form a 
vahiable check on the curve calculations and the instrumental 
work. 



CHAPTER II. 



ALIGNMENT. 



In this chapter the alignment of the center line only of a 
pair of rails is considered. When a railroad is crossing a sum- 
mit in the grade line, although the horizontal projection of the 
alignment may be straight, the vertical projection will consist of 
two sloping lines joined by a curve. When a curve is on a 
grade, the center line is really a spiral, a curve of double curva- 
ture, although its horizontal projection is a circle. The center 
line therefore consists of straight lines and curves of single 
and double curvature. The simplest method of treating them 
is to consider their horizontal and vertical projections separately. 
In treating simple, compound, and transition curves, only the 
horizontal projections of those curves will be considered. 



SIMPLE CURVES. 

1 8. Designation of curves. A curve may be designated either 
by its radius or by the angle subtended by a chord of unit 
length. Such an angle is known 
as the '' degree of curve " and is 
indicated by D. Since the curves 
that are practically used have very 
long radii, it is generally impracti- 
cable to make any use of the actual 
center, and the curve is located 
without reference to it. li AB in 
Fig. 4 represents a unit chord (C) 
of a curve of radius R, then by the 
above definition the angle AOB 
equals D, Then 




A0siniD = JA5 = iC. 



R = 



sin JD' 



(1) 



19 



20 



RAILROAD CONSTRUCTION. 



§19. 



or, by inversion, 



sin JD = 



2R' 



(2) 



The up.it chord is variously taken throughout the world as 
100 feet, 66 feet, and 20 meters. In the United States 100 
feet is invariably used as the unit chord length, and throughout 
this work it will be so considered. Table I has been computed 
on this basis. It gives the radius, with its logarithm, of all 
curves from a. 0° 01' curve up to a 10° curve, varying by single 
minutes. The sharper curves, which are seldom used, are given 
with larger intervals. 

An approximate value of R may be readily found from the 
following simple rule, which should be memorized: 



R'- 



5730 

D ' 



Although such values are not mathematically correct, since R 
does not strictly vary inversely as D, yet the resulting value is 
within a tenth of one per cent for all commonly used values 
of R, and is sufficiently close for many purposes, as will be 
shown later. 

19. Length of a subchord. Since it is impracticable to 
measure along a curved arc, curves are always measured by 

laying off 100-foot chord lengths. 
This means that the actual arc is 
always a little longer than the 
chord. It also means that a sub- 
chord (a chord shorter than the unit 
length) will be a little longer than 
the ratio of the angles subtended 
would call for. The truth of this 
may be seen without calculation 
by noting that two equal sub- 
chords, each subtending the angle 
Fig. 8. JD, will evidently be slightly longer 

than 50 feet each. If c be the length of a subchord subtend- 
ing the angle d, then, as in Eq. 2, 




sinir^ = ,^, 



§ 20. ALIGNMENT. 21 

or, by inversion, 

c = 2Rsinid (3) 

The nominal length of a subchord=100yr For example, 

a nominal subchord of 40 feet will subtend an angle of --^^^ of 
Z)°; its true length will be slightly more than 40 feet, and may 
be computed by Eq. 3. The difference between the nominal 
and true lengths is maximum when the subchord is about 57 
feet long, but with the low degrees of curvature ordinarily used 
the difference may be neglected. With a 10° curve and a 
nominal chord length of 60 feet, the true length is 60.049 feet. 
Very sharp curves should be laid off with 50-foot or even 25- 
foot chords (nominal length). In such cases especially the true 
lengths of these subchords should be computed and used instead 
of the nominal lengths. 

20. Length of a curve. The length of a curve is always 
indicated by the quotient of 100J-=-Z>. If the quotient of 
J-^D is a whole number, the length as thus indicated is the 
true length — measured in 100-foot chord lengths. If it is an 
odd number or if the curve begins and ends with a subchord 
(even though J-^-Z) is a whole number), theoretical accuracy 
requires that the true subchord lengths shall be used, although 
the difference may prove insigliificant. The length of the arc 
(or the mean length of the two rails) is therefore always in 
excess of the length as given above. Ordinarily the amount 
of this excess is of no practical importance. It simply adds an 
insignificant amount to the length of rail required. 

Example. Required the nominal and true lengths of a 
3° 45' curve having a central angle of 17° 25'. First reduce 
the degrees and minutes to decimals of a degree. (100 X 1 7° 25') 
4-3° 45'-174l.667 --3.75=464.444. The curve has four 100- 
foot chords and a nominal chord of 64.444 The true chord 
should be 64 451. The actual arc is 

17°.4167 X j|q5 X i^ =464.527 

The excess is therefore 464.527-464.451 =0.076 foot. 

21. Elements of a curve. Considering the line as running 
from A toward B, the beginning of the curve, at A, is called 
th'e point of curve {PC). The other end of the curve, at /?, U 



22 



RAILROAD CONSTRUCTION. 



§22. 



called the jpoint of tangency (PT). The intersection of the 

tangents is called the vertex (V). 
The angle made by the tangents 
at y, which equals the angle 
made by the radii to the extrem- 
ities of the curve, is called the 
central angle (J). AV and BVy 
the two equal tangents from the 
vertex to the PC and PT, are 
called the tangent distances (T). 
The chord AB is called the long 
chord (LC). The intercept HG 
from the middle of the long 
chord to the middle of the arc 
is called the middle ordinate (M). 
That part of the secant GV from 
the middle of the arc to the vertex is called the external distance 
(E). From the figure it is very easy to derive the following fre- 
quently used relations: 




Fig. 9. 



LC =2R sin i J 
M=R vers JJ 
E=R exsec Ji 



(4) 
(5) 
(6) 
(7) 



22. Relation between t, e, and J. Join A and G in Fig. 9. 
The angle FAG = J J, since it is measured by one half of the 
arc AG between the secant and tangent- AGO =90°— iJ. 

AV :VG :: sin AGV : sin VAG; 
sin AGV = sin AGO = cos iJ: 

T :E : icosiJ isinji; 

T=E cot IJ, . , . . (8) 

The same relation may be obtained by dividing Eq. 4 by Eq. 
7, since tan a — exsec a — cot Ja. 

23. Elements of a 1° curve. From Eqs. 1 to 8 it is seen that 
the elements of a curve vary directly as R. It is also seen to 
be very nearly true that R varies inversely as D. If the ele- 
ments of a 1° curve for various central angles are calculated and 
tabulated, the elements of a curve of Z>° curvature may be 
approximately found by dividing by D.the corresponding ele- 
ments of a 1° curve having the same central angle. For small 



I 



§24. 



ALIGNMENT. 



23 



central angles and low degrees of curvature the errors involved 
by the approximation are insignificant, and even for larger 
angles the errors are so small that for many purposes they may be 
disregarded 

In Table II is given the value of the tangent distances, 
external distances, and long chords for a 1° curve for various 
central angles The student should familiarize himself with the 
degree of approximation involved by solving a large number of 
cases under various conditions by the exact and by the approxi- 
mate methods, in order that he may know when the approxi- 
mate method is sufficiently exact for the intended purpose. 
The approximate method also gives a ready check on the 
exact method. 

24. Exercises, (a) What is the tangent distance of a 4° 20' 
curve having a central angle of 18° 24'? 

(b) Given a 3° 30' curve and a central angle of 16° 20', how 
far will the curve pass from the vertex? [Use Eq. 7.] 

(c) An 18° curve is to be laid off using 25-foot (nominal) 
chord lengths. What is the true length of the subchords? 

(d) Given two tangents making a central angle of 15° 24'. 
It is desired to connect these tangents by a curve which shall 
pass 16.2 feet from their intersection. How far down the 
tangent will the curve begin and what will be its radius? (Use 
Eq. 8 and then use Eq. 4 inverted.) 

25. Curve location by defiections. The angle between a 
secant and a tangent (or between two secants intersecting on an 
arc) is measured by one half of the intercepted arc. Beginning 
at the PC (A in Fig. 10), if the 
first chord is to be a full chord 
we may deflect an angle VAa 
(=^D), and the point a, which is 
100 feet from A, is a point on the 
curve. For the next station, b, 
deflect an additional angle hAa 
(=^D) and, with one end of the 
tape at a, swing the other end 
until the 100-foot point is on the 
line Ah. The point b is then on a ' 
the curve. If the final chord cB 
is a subchord, its additional deflec- Fig. 10. 

tion (id) is something less than JD. The last deflection (BAV) is 




24 RAILROAD CONSTRUCTION. § 26. 

of course Ji. It is particularly important, when a curve begins 
or ends with a subchord and the deflections are odd quantities, 
that the last additional deflection should be carefully com- 
puted and added to the previous deflection, to check the mathe- 
matical work by the agreement of this last computed deflec- 
tion with JJ. 

Example, Given a 3° 24' curve having a central angle of 
18° 22' and beginning at sta. 47 + 32, to compute the deflec- 
'tions, The nominal length of curve is 18° 22' ^3° 24' = 18.367 4- 
3.40 = 5.402 stations or 540.2 feet. The curve therefore ends 
at sta. 52 + 72.2. The deflection for sta. 48 is iVo-Xi(3° 24') 
= 0.68Xl°.7 = l°. 156 = 1° 09' nearly. For each additional 100 
feet it is 1° 42' additional. The final additional deflection for 
the final subchord of 72.2 feet is 

-^Xi(3° 24') =1°.2274 = 1° 14' nearly. 

The deflections are 

P. C . . . Sta. 47 + 32 0° 

48 0° +1°09' = 1°09' 

49 1° 09' + 1° 42' =2° 51' 

50 2° 51' + 1° 42' =4° 33' 

51 4° 33' + 1° 42' = 6° 15' 

52 6° 1 5' + 1° 42' = 7° 57' 

P. T 52 + 72.2 7°57' + l° 14'=9°11' 

As a check 9° ll' = K18° 22') =i^. (See the Form of Notes 
in § 17.) 

26. Instrumental work. It is generally impracticable to 
locate more than 500 to 600 feet of a curve from one station. 
Obstructions will sometimes require that the transit be mov^ed up 
every 200 or 300 feet. There are two methods of setting off 
the angles when the transit has been moved up from the PC. 

(a) The transit may be sighted at the previous transit station 
with a reading on the plates equal to the deflection angle from 
that station to the station occupied, but v/ith the angle set off on 
the other side of 0°, so that when the telescope is turned to 0° it 
will sight along the tangent at the station occupied. Plunging 
the telescope, the forward stations may be set off by deflecting 
the proper deflections from the tangent at the station occupied 



"^6 11 
ed I 



26. 



ALIGNMENT. 



25 



This is a very common method and, when the degree of curva- 
ture is an even number of degrees and when the transit is only 
set at even stations, there is but little objection to it. But the 
degree of curvature is sometimes an odd quantity, and the exi- 
gencies of difficult location frequently require that substations 
be occupied as transit stations. Method (a) will then require 
the recalculation of all deflections for each new station occupied. 
The mathematical work is largely increased and the probability 
of error is very greatly increased and not so easil}^ detected. 
Method (h) is just as simple as method (a) even for the most 
simple cases, and for the more difficult cases just referred to the 
superiority is very great. 

(b) Calculate the deflection for each station and substation 
throughout the curve as though the whole curve were to be lo- 
cated from the PC. The computations 
may thus be completed and checked (as 
above) before beginning the instrumental 
work. If it unexpectedly becomes neces- 
sary to introduce a substation at any 
point, its deflection from the PC may be 
readily interpolated. The stations actually 
set from the PC are located as usual. 
Rule. When the transit is set on any 
forward station, backsight to^NY previous 
station with the plates set at the deflection 
angle for the station sighted at. Plunge 
the telescope and sight at any forward 
station with the deflection angle originally 
computed for that station. When the 
plates read the deflection angle for the 
station occupied, the telescope is sighting 
along the tangent at that station — which 
is the method of getting the forward tan- 
gent when occupying the PT. Even though 
the station occupied is an unexpected sub- 
station, when the instrument is properly 
oriented at that station, the angle reading 
for any station, forward or back, is that originally computed 
for it from the PC. In difficult work, where there are ob- 
structions, a valuable check on the accuracy may be found by 
sighting backward at any visible station and noting whether 



Fig. 11. 



26 



KAILROAD CONSTRUCTION. 



§26. 



its deflection agrees with that originally computed. As a 
numerical illustration, assume a 4° curve, with 28° curvature, 
with stations 0, 2, 4, and 7 occupied. After setting stations 
1 and 2, set up the transit at sta. 2 and backsight to sta. 
with the deflection for sta. 0, which is 0°. The reading on sta. 
1 is 2°; when the reading is 4° the telescope is tangent to 
the curve, and when sighting at 3 and 4 the deflections will be 
6° and 8°. Occupy 4; sight to 2 with a reading of 4°. When 
the reading is 8° the telescope is tangent to the curve and, by 
plunging the telescope, 5, 6, and 7 may be located with the 
originally computed deflections of 10°, 12°, and 14°. When oc- 
cupying 7 a backsight may be taken to any visible station with 
the plates reading the deflection for that station; then when 





Fig. 12. 



Fig. 13. 



the plates read 14° the telescope will point along the forward 
tangent. 

The location of curves by deflection angles is the normal 
method. A few other methods, to be described, should be con- 
sidered as exceptional. 



§27. 



ALIGNMENT. 



27 



27. Curve location by two transits. A curve niight be located 
more or less on a swamp where accurate chaining would be 
exceedingly difficult if not impossible. The long chord AB 
(Fig. 12) may be determined by triangulation or otherwise, 
and the elements of the curve computed, including (possibly) 
subchords at each end. The deflection from A and B to each 
point may be computed. A rodman may then be sent (by 
whatever means) to locate long stakes at points determined 
by the simultaneous sightings of the two transits. 

28. Curve location by tangential offsets. When a curve is 
very flat and no transit is at hand the following method msiy be 
used (see Fig. 13) : Produce the back tangent as far forward as 
necessary. Compute the ordinates Oa\ Ob\ Oc% etc., and the 
abscissae a^a, h'h, c'c, etc. If Oa is a full station (100 feet), then 



Oa'-=Oa' =100 cos JD, also = i? sin D ; 

OW=Oa'^a'h' =100 cos ii) + 100 cos |D, 

also =i^ sin 2Z); 
Oc' = Oa' + a'V + 6 V = 100(cos \D + cos \D + cos |D) , 

also =jK sin 3Z>; 



etc. 



a' a = lOO^in JD, also =2^ vers D; 

6'6=a'a + &"6 =100 sin JZ) + 100 sin |Z), 

also=i^ vers 2Z); 
c'c^yh-\-c"c =100(siniD + sin|i)-fsin|D), 

also = J? vers 3Z>; 



(9) 



(10) 



etc. 

The functions JD, |D, etc., may be more conveniently used 
without logarithms, by adding the several natural trigonometrical 
functions and pointing off two decimal places. It may also be 
noted that Ob' (for example) is one half of the long chord for 
four stations; also that b'b is the middle ordinate for four 
stations. If the engineer is provided with tables giving the long 
chords and middle ordinates for various degrees of curvature, 
these quantities may be taken (perhaps by interpolation) from 
such tables. 

If the curve begins or ends at a substation, the angles and 
terms will be correspondingly altered, The modifications may 



28 RAILROAD CONSTRUCTION. § 29. 

be readily deduced on the same principles as above, and should 
be worked out as an exercise by the student. 

In Table II are given the long chords for a 1° curve for various 
values of J. Dividing the value as given b}^ the degree of the 
curve, we have an approximate value which is amply close for 
low degrees of curvature, especially for laying out curves with- 
out a transit. For example, given a 4° 30' curve, required the 
ordinate Oc\ This is evidently one half of a chord of six stations, 
with J =27°. Dividing 2675.1 (which is the long chord of a 
1° curve with J =27°) by 4.5 we have 594.47; one half of this is 
the required ordinate, Oc' =297.23. The exact value is 297.31, 
an excess of .08, or less than .03 of 1%. The true values 
are always slightly in excess of the value as computed from 
Table II. 

Exercise. A 3° 40' curve begins at sta. 18 + 70 and runs to 
sta. 23 + 60. Required the tangential offsets and their corre- 
sponding ordinates. The first ordinate = 30 cos KiVo X 3° 40') = 
30 X. 99995 =29.9985; the offset =30 sin 0° 33' = 30 X. 0096 = 
0.288. For the second full station (sta. 20) the ordinate = 
i long chord for i=2(l° 06' + 3° 40') with 7) = 3°40'. Divid- 
ing 476.12, from Table II, by 3|, we have 129.85. Otherwise, 
by Eq. 9, the ordinate = 30 X cos 0° 33' + 100 cos (1° 06' + 1° 50') 
= 30.00 + 99.87 = 129.87. The offset for sta. 20=30 sin 0° 33' + 
100 sin (1° 06' + 1° 50') = 0.288 + 5.12 =5.41. Workout 
similarly the ordinates and offsets for sta. 21, 22, 23, and 
23 + 60. 

29. Curve location by middla ordinatss. Take first the sim- 
pler case when the curve begins at an even station. If we con- 
sider (in Fig. 14) the curve produced back to z, the chord za — 
2X100 cos iZ), A 'a = 100 cos JD, and A'A=^am=zn = 100 sin JD.- 
Set off A A' perpendicular to the tangent and A 'a parallel to 
the tangent. AA' =aa' =66' =cc', etc. = 100 sin JZ). Set off 
aa' perpendicular to a^A. Produce Aa^ until a'b=A^a, thus 
determining h. Succeeding points of the curve may thus be 
determined indefinitely. 

Suppose the curve begins with a subchord. As before 
ra = Am' =c^ cos J<i', and rA =am^ =c^ sin ^d\ Also sz = An^ = 
c"cosK', and %A=2yi,'=c"sinid", in which ((f+c?")=Z). 
The points z and a being determined on the ground, aa' may 
be computed and set off as before and the curve continued in 



§ 30. 



ALIGNMENT. 



29 



full stations. A subchord at the end of the curve may be located 
by a similar process. 

30. Curve location by offsets from the long chord. (Fig. 16.) 
Consider at once the general case in which the curve commences 
with a subchord (curvature, d')j continues with one or more full 





Fig. 14. 



Fig. 15, 



Fig. ]6. 



chords (curvature of each, D), and ends with a subchord with 
curvature d". The numerical work consists in computing first 
ABy then the various abscissae and ordinates. AB=2R sin §J. 



Aa'^Aa' —a' cos^{A—d')\ 

Ab'^Aa' + a'b' =c' cos K^ -<^') + 100 cos K^ -2c^'-7)); 

Ac=Aa'-\-a'h'-\-Vc^ = c' cos ^(J-c?0 + 100 cos K^ -2rf'-Z)) 

+ 100cosi(J-2rf"-£)); 
also 

-=AB-B<^ =2S sin \A-cr cos \{A-6r^, 



(11) 



a'a = a'a =c' sin ^(i — cfO; 

h'h = a'a + m'b = c' sin i(i -6^') + 100 sin ^( J -2c?'- D); 

r'c =6'6-n6 = c' sin K^-c?') + 100 sin K^-2rf'-D) \ (12) 

-100sin^(i-2d''-Z)); 
also =c" sin ^{d — d")» 



The above formulae are considerably simphfied when the 



30 



RAILROAD CONSTRUCTION. 



§31. 



curve begins and ends at even stations. When the curve is 
very long a regular law becomes very apparent in the formation 
of all terms between the first and last. There are too few terms 
in the above equations to show the law. 

31. Use and value of the above methods. The chief value 
of the above methods lies in the possibility of doing the work 
without a transit. The same principles are sometimes em- 
ployed, even when a transit is used, when obstacles prevent the 
use of the normal method (see § 32, c). If the terminal tan- 
gents have already been accurately determined, these methods 
are useful to locate points of the curve when rigid accuracy is 
not essential. Track foremen frequently use such methods to 
lay out unimportant sidings, especially when the engineer and 
his transit are not at hand. Location by tangential offsets (or 
by offsets from the long chord) is to be preferred when the 
curve is fiat (i.e., has a small central angle J) and there is no 
obstruction along the tangent, or long chord. Location by 
middle ordinates may be employed regardless of the length of 
the curve, and in cases when both the tangents and the long 
chord are obstructed. The above methods are but samples 
of a large number of similar methods which have been devised. 
The choice of the particular method to be adopted must be 
determined by the local conditions. 

32. Obstacles to location. In this section will be given only 
a few of the principles involved in this 
class of problems, with illustrations. The 
engineer must decide, in each case, which 
is the best method to use. It is frequently 
advisable to devise a special solution for 
some particular case. 

a. When the vertex is inaccessible. As 
shown in § 26, it is not absolutely essential 
that the vertex of a curve should be 
located on the ground. But it is very evi- 
dent that the angle between the terminal 
tangents is determined with far less prob- 
able error if it is measured by a single 
measurement at the vertex rather than as 
the result of numerous angle measurements 
Fig. 17. along the curve, involving several posi- 

tions of the transit and comparatively short sights. Some- 




§ 32. ALIGNMENT. 31 

times the location of the tangents is already determined on 
the ground (as by hn and am, Fig. 17), and it is required to 
join the tangents by a curve of given radius. Method. Measure 
ah and the angles Vha and haV, A is the sum of these angles. 
The distances hV and aV are computable from the above data. 
Given A and R, the tangent distances are computable, and then 
Bh and a A are found by subtracting hV and aV from the tan- 
gent distances. The curve may then be run from A, and the 
work may be checked by noting whether the curve as run ends 
at B — previously located from h. 

Example. Assume a?>=546 82; angle a = 15° 18'; angle 
h = 18° 22' ; Z) = 3° 40' ; required aA and hB. 
J = 15° 18' + 18° 22' =33° 40' 

Eq. (4) R (3° 40') 3.19392 

tanlJ=tanl6°50' 9.4808 

r=472.85 2.67475 

^^ , sin 18° 22' ah 2.73784 

^^ ■" sm 33° 40' log sin 18° 22' 9.49844 

co-log sin 33° 40' 0.25621 

ay=310.81 2.49250 

A 7=472. 85 

aA =162.04 



^ sin 33° 40' log sin 15° 18' 9.42139 

co-log sin 33° 40' 0.25621 

6F=260.29 2.41545 

5F=472.85 

6^=212.56 



b. When the point of curve (or point of tangency) is inacces- 
sible, i^t some distance (As, Fig. 18) an unobstructed line jpn 
may be run parallel with AV, nv=py=As=R vers a, 

\ vers a=As-7-R, 

ns=ps=R sin a^ 



32 



KAILROAD CONSTRUCTION. 



33. 



At 2/, which is at a distance ps back from the computed posi- 
tion of A J make an offset sA 
to p. Run pn parallel to the 
tangent. A tangent to the 
curve at n makes an angle of a 
with np. From n the curve is 
run in as usual 

If the point of tangency is 
obstructed, a similar process, 
somewhat reversed, may be 
used. 13 is that portion of J still 
to be laid off when m is reached. 
tm=tl=R sin 13. mz=tB=lx=R 
vers B. 

c. When the central part of 

the curve is obstructed, a is the 

central angle between two points 

of the curve between which 

a may equal any angle, but it is prefer- 




Fig. 18. 



a chord may be run 
able that a should bo a multiple 
of D, the degree of curve, and that 
the points m and n should be on 
even stations. mn=2Rs\n^a. A 
point s may be located by an offset 
hs from the chord mn by a similar 
method to that outlined in § 30. 

The device of introducing the 
dotted curve mn having the same 
radius of curvature as the other, 
although neither necessary nor 
advisable in the case shown in 
Fig. 19, is sometimes the best 
method of surveying around an 
obstacle. The offset from any point on the dotted curve to 
the corresponding point on the true curve is twice the ^' ordinate 
to the long chord,'' as computed in § 30. 

33. Modifications of location. The following methods may 
be used in allowing for the discrepancies between the ''paper 
location'' based on a more or less rough preliminary survey and 
the more accurate instrumental location. (See § 15.) They are 




II 



§33. 



ALIGNMENT. 



33 



also frequently used in locating new parallel tracks and modify- 
ing old tracks. 

a. To move the forward tangent parallel to itself a distance a?, 
the point of curve (^4) remaining fixed. (Fig. 20.) 



VV' = 






sinhVV^ sin i' 
AV'=AV-\-VV\ 
The triangle BmB^ is isosceles and Bm-=B'm, 

B'r x' 



R'-R=0'0==mB = 



vers B^mB vers A' 



(13) 



;. R'=R + 



X' 



vers J* 



(14) 



The solution is very similar in case the tangent is moved in- 
ward to Y"B" , Note that this method necessarily changes the 




Fig. 20. 




Fig. 21. 



radius. If the radius is not to be changed, the point of curve 
must be altered as follows: 

b. To move the forward tangent parallel to itself a distance a?, 
the radius being unchanged. (Fig. 21.) In this case the whole 



34 



RAILROAD CONSTRUCTION. 



§34. 



curve is moved bodily a distance 00^ - 
moved parallel to the first tangent AV 

B'n ^ X 
sin nBB^ sin J 



=AA' = VV'==BB% and 



BB' = -r^ 



-AA' 



(15) 



c. To change the direction of the forward tangent at the point 
of tangency. (Fig. 22.) This problem involves a change (a) in 

the central angle and also requires a 
new radius. An error in the deter- 
mination of the central angle fur* 
nishes an occasion for its use. 
R, Ay a, AVy and BV are known. 




Bs=R vers A, 



R'=R- 



Bs^R'veTsA', 
vers A 



(16) 



"vers {A — a)' 
Fig. 22. ' ^s=i^ sin i. A's=R' sin A'. 

.\ AA'=A's'-As==R' sin A' -R sin A. . . (17) 

The above solutions are given to illustrate a large class of 
problems which are constantly arising. All of the ordinary 
problems can be solved by the application of elementary geome- 
try and trigonometry. 

34. Limitations in location. It may be required to run a 
curve that shall join two given tangents and also pass through a 
given point The point (P, Fig. 
23) is assumed to be deter- 
mined by its distance (VP) 
from the vertex and by the 
angle AFP=/?. 

It is required to determine 
the radius (R) and the tangent 
distance (AV). A is known. 
PVG = i(180''- A) -1^ 
=90°-(ii+/?). 
PP' =2VP sin PVG 

-2yPcos(ii+/?). 
PSV^iA. 




SP^VP-$^. 
sm ^A 



Fig. 23. 



§ 35. ALIGNMENT. 35 

JVP4^[VP^-,2VP cos (iJ+/?)1 



_VP I sin^^ 2sin/?cos(^i+/9) 
""'^^^^sinn^ sin J J 

sin J J 

= g-^[sin(ii+/?) + Vsin2/? + 2sin/?sinJJcos(JJ+/?)]. (18) 

i^=AFcotiJ. 

In the special case in which P is on the median Hne OV, 
^=90°- JJ, and (iJ+^)=90^ Eq. 18 then reduces to 

2lF=4-^Xl + COS y) ^VP cot iJ, 

as might have been immediately derived from Eq. 8. 

In case the point P is given by the offset PK and by the 
distance VK^ the triangle PKV may be readily solved, giving the 
distance VP and the angle /?, and the remainder of the solution 
will be as above. 

35. Determination of the curvature of existing track, (a) Using 
a transit. Set up the transit at any point in the center of the 
track. Measure in each direction 100 feet to points also in the 
center of the track. Sight on one point with the plates at 0°. 
Plunge the telescope and sight at the other point. The angle 
between the chords equals the degree of curvature. 

(b) Using a tape and string. Stretch a string (say 50 feet 
long) between two points on the inside of the head of the outer 
rail. Measure the ordinate {x) between the middle of the string 
and the head of the rail. Then 

„ chord^ . , ^ 

^=—g^( very nearly) (19) 

For, in Fig. 24, since the triangles AOE and ADC are similar, 




36 RAILROAD CONSTRUCTION^ § 36. 

AO:AE::AD:DC or R = ^AD''-^x. When, as is usual, 
the arc is very short compared with the 
radius, AD = ^ABj very nearly. Making 
this substitution we have Eq. 19. With a 
chord of 50 feet and a 10° curve, the result- 
ing difference in x is .0025 of an inch — far 
within the possible accuracy of such a 
method. The above method gives the 
radius of the inner head of the outer rail. 
It should be diminished by \g for the radius 
of the center of the track. With easy curvature, however, this 
will not affect the res 'j It by more than one or two tenths of one 
per cent. 

The inversion of this formula gives the required middle or- 
dinate for a rail on a given curve. For example, the middle 
ordinate of a 30-foot rail, bent for a 6° curve, is 

x=900--(8X955) =.118 foot = 1.4 inches. 

Another much used rule is to require the foreman to have a 
string, knotted at the center, of such length that the middle 
ordinate, measured in inches, equals the degree of curve. To 
find that length, substitute (in Eq. 19) 5730 -f-D for R and 
D-^12 for X. Solving for chord, we obtain chord = Q\.S feet. 
The rule is not theoretically exact, but, considering the uncertain 
stretching of the string, the error is insignificant. In fact, the 
distance usually given is 62 feet, which is close enough for all 
purposes for which such a method should be used. 

36. Problems. A systematic method of setting down the 
solution of a problem simplifies the work. Logarithms should 
always be used, and all the work should be so set down that a 
revision of the work to find a supposed error may be readily 
done. The value of such systematic work will become more 
apparent as the problems become more complicated. The two 
solutions given below will illustrate such work. 

a. Given a 3° curve beginning at Sta. 27-f 60 and running 
to Sta. 32 + 45. Compute the ordinates and offsets used in 
locating the curve by tangential offsets. 

h. With the same data as above, compute the distances to 
locate the curve by offsets from the long chord. 

C, Assume that in Fig. 17 ah is measured as 217.6 feet, the 



§ 36. ALIGNMENT. 37 

angle a?>y = 17°42', and the angle haV = 21°U\ Join the 
tangents by a 4° 30' curve. Determine bB and aA. 

d. Assume that in a case similar to Fig. 18 it was noted 
that a distance (As) equal to 12 feet would clear the building. 
Assume that i=38°20' and that i)=4°40'. Required the 
value of a and the position of n. Solution: 

vers a=ils-v-i^ As = 12 log = l. 07918 

R (for 4° 40' curve) log = 3.08923 

a^°Or log vers a = 7.98991 

ns=i^ sin a log sin a =9. 14445 

log E = 3. 08923 
715 = 171.27 log =2. 23369 

c. Assume that the forward tangent of a 3° 20' curve having 
a central angle of 16° 50' must be moved 3.62 feet inward, with- 
out altering the P.C. Required the change in radius. 

/. Given two. tangents making an angle of 36° 18'. It is 
required to pass a curve through a point 93.2 feet from the 
vertex, the line from the vertex to the point making an angle 
of 42" 21' with the tangent. Required the radius and tangent 
distanct. Solution: Applying Eq. 18; we have 

2 log= 0.30103 

,5 = 42° 21' log sin = 9.82844 

ij = 18°09' log sin = 9.49346 

(1 i +;9) =60° 30' log cos = 9.69234 

.20667 9.31527 

log sm2 /? =9 . 65688 .... .453 82 " 

^|9. 81987 .66049 

9.90993 'T8T27i 

nat. sin 60° 30' .8703 

1.6830 log= 0.22610 

FP = 93.2 log = 1.96941 

2.19551 
log sin JJ= 9.49346 

Tang, dist. ^F = 503.36 Jog= 2.70205 

log cot i J = 10.48437 

7^ = 1536.1 log= 3.18642 

i)=3°44' 



38 



RAILKOAD CONSTRUCTION. 



§37. 



COMPOUND CURVES. 

37. Nature and use. Compound curves are formed by a 
succession of two or more simple curves of different curvature. 
The curves must have a common tangent at the point of com- 
pound curvature (P.C.C). In mountainous regions there is 
frequently a necessity for compound curves having several 
changes of curvature. Such curves may be located separately 
as a succession of simple curves, but a combination of two 
simple curves has special properties which are worth investigat- 
ing and utilizing. In the following demonstrations R2 always 
represents the longer radius and Ri the shorter j no matter which 
succeeds the other. T^ is the tangent adjacent to the curve of 
shorter radius (Ri), and is invariably the shorter tangent. J^ is 
the central angle of the curve of radius Ri, but it may be greater 
or less than Jg 

38. Mutual relations of the parts of a compound curve having 
two branches. In Fig. 25, AC and CB are the two branches of 




Fig. 25. 



^1 



the compound curve having radii of Ri and R2 and central angles 
of ii and ^2- Produce the arc AC to n so that AO^ri^d. The 
chord Cn produced must intersect B. The line ns, parallel to 
CO2, will intersect BO^ so that Bs = sn = 020^=R2—Rv Draw 
Am perpendicular to O^a, It will be parallel to hk. 



§ 38. ALIGNMENT. 39 

Br = sn vers Bsn = (R2 — Ri) vers J 2 ; 
mn=AOi vers AO{n =R^ vers J; 
^/c=^7sinAF/c =Tisiii J; 

.-. Tisin i=J?i vers i + (E2-^i) vers ^2- • . (20) 
Similarly it may be shown that 

T2 sin A =R2 vers J-(i?2-^i) vers ij. » . (21) 

The mutual relations of the elements of compound curves 
may be solved by these two equations. For example, assume 
the tangents as fixed (i therefore known) and that a curve of 
given radius Ri shall start from a given point at a distance T^ 
from the vertex, and that the curve shall continue through a 
given angle J^. Required the other parts of the curve. From 
Eq. 20 we have 

Ti sin J—Ri vers J 



R2 — Ri^ 



vers Jo 



... R^=R^ + Tl^^A^p]hj^ (22) 

vers(J — ii) 

T2 may then be obtained from Eq. 21. 

As another problem, given the location of the two tangents, 
with the two tangent distances (thereby locating the PC and 
PT), and the central angle of each curve; required the two 
radii. Solving Eq. 20 for R^, we have 

j^ _Ti sin J—R2 vers J2 
^ vers J— vers J 2 

Similarly from Eq. 21 we may derive 

T2 sin J — i?2(vers J —vers J.) 

Ki = -. . 

vers Jj 

Equating these, reducing, and solving for R^, we have 

Ti sin J vers J J — T2 sin J (vers J — vers z/2) 
'^~ vers A2 vers Jj — (vers J— vers Ji)(vers J— vers A^' ^ 

Although the various elements may be chosen as above with 

considerable freedom, there are limitations. For example, in 

Eq. 22, since R2 is always greater than R^, the term to be 

added to R^ must be essentially positive — i.e., T^ sin A must be 

vers A J 

greater than R^ vers A. This means that T{>R^— — j-, or that 



40 



EAII.EOAD CONSTRUCTION. 



§39. 



Ti>7?itanji, or that T^ is greater than the corresponding 
tangent on a simple curve. Similarly it may be shown that Tg 
is less than R2 tan §J or less than the corresponding tangent 
on a simple curve. Nevertheless T2 is always greater than T^. 
In the limiting case when i?2=^i> ^2 = ^1^ ^^^ ^2 = ^1- 

39. Modifications of location. Some of these modifications 
may be solved by the methods used for simple curves. For 
example ; 

a. It is desired to move the tangent VB, Fig. 26, parallel to 
itself to VB' . Run a new curve from the P.C.C. which shall 
reach the new tangent at B\ where the chord of the old curve 





Fig. 26. 



Fig. 27. 



intersects the new tangent. The solution is almost identical 
w^ith that in § 33, a. 

b. Assume that it is desired to change the forward tangent 
(as above) but to retain the same radius. In Fig. 27 

{R2 — R1) cos J 2 =^2^ J 

lR2-Ri) cos J 2' =0^n\ 



x = 02n — 02n' ={R2 — Ri)(cos ^2 — <^os J 2^). 

X 



cos J/ = COS J2—T> D 
it2— iXi 



(24) 



The P.C.C. is moved backioard along the sharper curve an 
angular distance of J2'~'^2 = ^i~^/- 

In case the tangent is moved inward rather than outward, 
the solution will apply by transposing J2 and ^2'- Then we 
§hall have 

cos J/ = cos J2 + 15 D 



(25) 



39. 



ALIGNMENT. 



41 



The P.C.C. is then moved forward. 

c. Assume the same case as (b) except that the larger radius 
comes first and that the tangent adjacent to the smaller radius 
is moved. In Fig. 28 

(R2-R1) cos J, =0{a\ 
(i^2-Ei)eosi/=0iV. 

x=0^^n' —0{ri 
= (R2—Ri)(cos J/— cos ii). 



cos J/ = cos ii + ^- 



-Ji: 



(26) 



The P.C.C. is moved forward 

along the easier curve an angular 

distance of J/ — il = i2~^2'' 
In case the tangent is moved inward, transpose as before and 

we have 




cos i/ = cos Ji — 



R2—R1 



(27) 



The P.C.C. is moved backward 

d. Assume that the radius-of one curve is to be altered with- 
out changing either tangent. Assume conditions as in Fig. 29. 

For the diagrammatic solution 
assume that R2 is to be increased 
by O2S. Then, since J?,' must 
pass through 0^ and extend be- 
yond Oj a distance O^S, the 
locus of the new center must lie 
on the arc drawn about 0^ as 
center and wnth OS as radius. 
The locus of O2 is also given 
by a line 0/p parallel to BV 
and at a distance of R2 (equal 
to S .. . P.C.C.) from it. The 
new center is therefore at the 
intersection Oo'. An arc with ra- 




FiG. 29. 
curve produced at new P.C.C. 



dius 7?2' will therefore be tangent 
at B^ and tangent to the old 
Draw 0,n^ perpendicular to O2B, 



42 RAILROAD CONSTRUCTION. § 40. 

With O2 as center draw the are 0{m, and with O2' as center draw 
the arc 0{m\ mB=m'B' =R^, 

.*. mn =m'n' = {R2 —Ri) vers J^/ = (R2 —Ri) vers ^2- 

/. vers i/ = ^^^^ vers ^2 (28) 

0{n = (R2—Ri) sin J2J 
Oin' = (i?2'-A)sini2'. 
BB' = 0{a' -0{ii = {R2' -Ri) sin A2' -{R2-R1) sin J2. (29} 

This problem may be further modified by assuming that the 
radius of the curve is decreased rather than increased, or that 
the smaller radius follows the larger. The solution" is similar 
and is suggested as a profitable exercise. 

It might also be assumed that, instead of making a given 
change in the radius R2, a given change BB^ is to be made. A2 
and R2 are required. Eliminate R2 from Eqs. 28 and 29 
and solve the resulting equation for A2. Then determine R2 
by a suitable inversion of either Eq. 28 or 29. 

As in §§ 32 and 33, the above problems are but a few, although 
perhaps the most common, of the problems the engineer may 
meet with in compound curves. All of the ordinary problems 
may be solved by these and similar methods. 

40. Problems, a. Assume that the two tangents of a com- 
pound curve are to be 348 feet and 624 feet, and that ij =22° 16' 
and ^2=28° 20'. Required the radii. 

[Arts, El =326.92; i?2 = 1574.85.] 

6. A line crosses a valley by a compound curve which is first 
a 6° curve for 46° 30' and then a 9° 30' curve for 84° 16'. It is 
afterward decided that the last tangent should be 6 feet farther 
up the hill. What are the required changes? {Note. The 
second tangent is evidently moved outward. The solution cor- 
responds to that in the first part of § 39, c. The P.C.C. is 
moved forward 16.39 feet. If it is desired to know how far the 
P.T. is moved in the direction of the tangent (i.e., the projection 
of BB\ Fig. 28, on V'B'), it may be found by observing that it 
is equal to nn' = (J?2~-^i)(sin ^1— sin J/). In this case it equals 
0.65 foot, which is very small because J^ is nearly 90°. The 
value of J2 (46° 30') is not used, since the solution is independent 
of the value of Jg* The student should learn to recognize 



§ 41. ALIGNMENT. 43 

which quantities are mutually related and therefore essential 
to a solution, and which are independent and non-essential.J 

TRANSITION CURVES. 

41. Superelevation of the outer rail on curves. When a mass 
is moved in a circular path it requires a centripetal force to keep 
it moving in that path. By the principles of mechanics we I 
know that this force equals Gv'^-^gR, in w^hich G is the weight, 
V the velocity in feet per second, g the acceleration of gravity in 
feet per second in a second, and R the radius of curvature. 
If the two rails of a curved track were laid on a level (trans- 
versely), this centripetal force could only be furnished by the 
pressure of the wheel-flanges against the rails. As this is very 
objectionable, the outer rail is elevated so that the reaction of 
the rails against the wheels shall 
contain a horizontal component 
equal to the required centripetal 
force. In Fig. 30, if oh represents 
the reaction, oc will represent the 
weight G, and ao wdll represent the 
required centripetal force. From •— — \ T-i---' — I"' 
similar triangles we may- write m^^ \ &•; 

sn : sm :\ ao : oc. Call g = 32.17. V S ' 

Call i^ = 5730-T-D, which is suffi- 
ciently accurate for this purpose (see 

§ 19). CaU ^ =52807-^3600, in which V is the velocity in miles 
per hour, mn is the distance between rail centers, which, for 
an 80-lb. rail and standard gauge, is 4.916 feet sm is slightly 
less than this. As an average value we may call it 4.900, which 
is its exact value when the superelevation is 4| inches. Calling 
sn=e, measured in feet, we have 

e=sm-=4.9^'i= 4.9X5280^72^ 



oc ' gR G 32.17X36002X5730'* 
e = . 0000572 F^i) (30) 

It should be noticed that, according to this formula, the re- 
quired superelevation varies as the square of the velocity, which 
means that a change of velocity of only 10% would call for a 
change of superelevation of 21%. Since the velocities of trains 
over any road are extremely variable, it is impossible to adopt 



44 



EAILKOAD CONSTRUCTION. 



§42. 



any superelevation which will fit all velocities even approx- 
imately. The above fact also shows why any over-iefinement 
in the calculations is useless and why the above approximations, 
which are really small, are amply justifiable. For example, the 
above formula contains the approximation that R = 5730^D. 
In the extreme case of a 10° curve the error involved w^ould be 
about 1%. A change of about J of 1% in the velocity, or say 
from 40 to 40.2 miles per hour, would mean as much. The error 
in e due to the assumed constant value of sm is never more than 
a very small fraction of 1%. The rail-laying is not done closer 
than this The following tabular form is based on Eq. (30) : 

SUPERELEVATION OF THE OUTER RAIL (iN FEET) FOR VARIOUS 
VELOCITIES AND DEGREES OF CURVATURE. 



Velocity in 
Miles per 


Degree of Curve. 


Hour. 


1° 


2° 


3° 


4^ 

.20 
.37 

,.57 
.82 


5° 


6° 


7° 


8° 


9° 


10° 


30 


.05 
.09 
.14 
.20 


.10 
.18 
.29 

.41 


.15 
.27 

_43__ 
i .62 


.26 

.71 


.31 

.86 


.36 


.41 


.46 


1.51 


40 
50 
60 


.64 


.73 


.82 





42. Practical rules for superelevation. A much used rule for 
superelevation is to '' elevate one half an inch for each degree of 
curvature. '^ The rule is rational in that e in Eq. 30 varies 
directly as D. The above rule therefore agrees with Eq. 30 
when V is about 27 miles per hour. However applicable the 
rule may have been in the days of low velocities, the elevation 
thus computed is too small now. The rule to elevate one inch 
for each degree of curvature is also used and is precisely similar 
in its nature to the above rule. It agrees with Eq. 30 when 
the velocity is about 38 miles per hour, which is more nearly 
the average speed of trains. 

Another (and better) rule is to ^^ elevate for the speed of the 
fastest trains. '^ This rule is further justified by the fact that a 
four-wheeled truck, having two parallel axles, will always tend 
to run to the outer rail and will require considerable flange pres- 
sure to guide it along the curve. The effect of an excess of super- 
elevation on the slower trains will only be to relieve this flange 
pressure somewhat. This rule is coupled with the limitation 



§42. 



ALIGNMENT. 



45 



that the elevation should never exceed a limit of six inches — 
sometimes eight inches. This limitation implies that locomo- 
tive engineers must reduce the speed of fast trains around sharp 
curves until the speed does not exceed that for which the actual 
superelevation used is suitable. The heavy line in the tabular 
form (§41) shovv^s the six-inch limitation. 

Some roads furnish their track foremen with a list of the super- 
elevations to be used on each curve in their sections. This 
method has the advantage that each location may be separately 
studied, and the proper velocity, as affected by local conditions 
(e.g., proximity to a stopping-place for all trains), may be de- 
termined and applied. 

Another method is to allow the foremen to determine the 
superelevation for each curve by a simple measurement taken 
at the curve. The rule is developed as follows: By an inversion 
of Eq. 19 we have 

x=chord^---8R, .,.•.. (31) 

Putting X equal to e In Eq. 30 and soMng for *' chord/' we 
have 

chord 2 = .0000572 F22)rj2 

=2.62172. 
chord = 1.Q2V. ........ (32) 

To apply the rule, assume that 50 miles per hour is fixed as 
the velocity from which the superelevation is to be computed. 
Then 1.627 = 1.62X50 = 81 feet, which is the distance given to 
the trackmen. Stretch a tape (or even a string) with a length 
of 81 feet between two points on the concave side of the head of 
cither the inner or the outer rail. The ordinate at the middle 
point then equals the superelevation. The values of this chord 
length for varying velocities are given in the accompanying 
tabular form. 



Velocity in miles per hour. . 
Chord length in feet , 



20 


25 


30 


35 


40 


45 


50 


55 


32.4 


40.5 


48.6 


56.7 


64.8 


72.9 


81.0 


89.1 



60 
97.2 



The following tabular form shows the standard (at one time) 

I on the N. Y., N. H. & H. R. R. It should be noted that the 

I elevations do not increase proportionately with the radius, and 

that they are higher for descending grades than for level or 



46 



BAILEOAB CONSTEUCTION. 



§43. 



ascending grades. This is on the basis that the velocity on curves 
and on ascending grades will be less than on descending grades. 
For example, the superelevation for a 0° 30' curve on a de- 
scending grade corresponds to a velocity of about 54 miles per 
hour, while for a 4° curve on a level or ascending grade the super- 
elevation corresponds to a velocity of only about 38 miles per 
hour. 



TABLE OF THE SUPERELEVATION OF THE OUTER RAIL ON CURVES. 
N. Y., N. H. & H. R. R. 



Degree of 


Level or as- 


Descending 


curve. 


cending grade. 


grade. 




inches. 


inches. 


0° 30' 


Of 


1 


1 00 


H 


If 


1 15 


If 


2 


1 30 


2 


2i 


1 45 


2i 


2i 


2 00 


2| 


2f 


2 15 


2f 


3 


2 30 


21 


3i 


2 45 


3 


31 


3 00 


3i 


31- 


3 15 


3i 


Si 


3 30 


3f 


4 


3 45 


Si 


4i 


4 00 


4 


H 



43. Transition from level to inclined track. On curves the 
track is inclined transversely ; on tangents it is level. The tran- 
sition from one condition to the other must be made gradually. 
If there is no transition curve, there must be either inclined 
track on the tangent or insufficiently inchned track on the curve 
or both. Sometimes the full superelevation is continued through 
the total length of the curve and the ^'run-off" (having a length 
of 100 to 400 feet) is located entirely on the tangents at each 
end. In other practice it is located partly on the tangent and 
partly on the curve. Whatever the method, the superelevation 
is correct at only one point of the run-off. At all other points 
it is too great or too small. This (and other causes) produces 
objectionable lurches and resistances when entering and leav- 
ing curves. The object of transition curves is to obviate these 
resistances. 

On the liChigh Valley R. R, the run-off is made in the form 
of a reversed vertical curve, as shown in the accompanying 
figure. According to this system the length of run-off varies 



§44. 



ALIGNMENT. 



47 



from 120 feet, for a superelevation of one inch, to 450 feet, 
for a superelevation of ten inches. Such a superelevation 
as ten inches is very unusual practice, but is successfully 
operated on that road. The curve is concave upward for two- 
thirds of its length and then reverses so that it is convex upward. 

TABLE FOR RUN-OFF OF ELEVATION OF OUTER RAIL OF CURVES. 
Drop in inches for each 30-foot rail commencing at theoretical point of curve. 



r 

2" 
4" 


V 

30 
30 


Y 

30 
30 
30 
30 
30 
30 
30 
30 


¥ 
30 

30 


Y 

30 
30 
30 
30 
30 
30 


F 
30 


Y 

30 
30 

30 
30 


¥ 

30 

30 
30 


\" 


W 


W 


IF 


\" 


¥ 

30 
30 

30 
30 


r 

30 
30 
30 
30 
30 
30 


¥ 

30 
30 
30 
30 
30 
30 
30 
30 


¥ 

30 

30 
30 
30 

30 
30 


¥ 

30 
30 

30 
30 
30 
30 


V 

30 

30 
30 
30 
30 


A" 

'30 
30 
30 


¥ 

30 
30 
30 
30 
30 
30 
30 


'30 
30 
30 


|_ 

120 
150 
180 
*?40 


^" 


30 

30 
30 








30 
30 
30 
30 
30 
30 


'^ro 


6'' 

V 

10'' 


30 
30 
30 
30 
30 


30 
30 


'30 
30 
30 
30 


300 
330 
360 
420 
450 



Drop 




The figure (and also the lower line of the tabulated form) 
shows the drop for each thirty-foot rail length. For shorter 
lengths of run-off, the drop for each 30 feet is shown b}^ the cor- 
responding lines in the tabular form. Note in each horizontal 
line that the sum of the drops, under which 30 is found, equals 
the total superelevation as found in the first column. For 
example, for 4 inches superelevation, length of curve 240 feet, 
the successive drops are \" , V' , y , y] %" , y, y, and y 
whose sum is 4 inches. Possibly the more convenient form 
would be to indicate for each 30-foot point the actual super- 
elevation of the outer rail, which w^ould be for the above case 
(running from the tangent to the curve) J'', f, J", IV', 2|", 



44. Fundamental principle of transition curves. If a curve 



48 RAILROAD CONSTRUCTION. § 45. 

has variable curvature, beginning at the tangent with a curve 
of infinite radius, and the curvature gradually sharpens until it 
equals the curvature of the required simple curve and there 
becomes tangent to it, the superelevation of such a transition 
curve may begin at zero at the tangent, gradually increase to 
the required superelevation for the simple curve, and yet have 
at every point the superelevation required by the curvature at 
jthat point. Since in Eq. (30) e is directly proportional to Z), 
the required curve must be one in which the degree of curve 
increases directly as the distance along the curve. The mathe- 
matical development of such a curve is quite complicated. It 
has, however, been developed, and tables have been computed for 
its use, by Prof. C. L. Crandall. The following method has the 
advantage of great simplicity, while its agreement with the true 
transition curve is as close as need be, as will be shown. 

45. Multiform compound curves. If the transition curve com- 
mences with a very flat curve and at regular even chord lengths 
compounds into a curve of sharper curvature until the desired 
curvature is reached, the increase in curvature at each chord 
point being uniform, it is plain that such a curve is a close ap- 
proximation to the true spiral, especially since the rails as laid 
will gradually change their curvature rather than maintain a 
uniform curvature throughout each chord length and then 
abruptly change the curvature at the chord points. Such a 
curve, as actually laidj will be a much closer approximation 
to the true curve than the multiform compound curve by which 
it is set out. There will actually be a gradual increase in curva- 
ture which increases directly as the length of the curve. 

46. Required length of spiral. The required length of spiral 
evidently depends on the amount of superelevation to be gained, 
*and also depends somewhat on the speed. If the spiral is laid 
off in 25-foot chord lengths, with the first chord subtending a 1° 
curve, the second a 2° curve, etc., the fifth chord will subtend 
a 5° curve, and the increase from this last chord to a 6° curve 
is the same as the uniform increase of curvature between the 
chords. The same spiral extended would run on to a 12° curve 
in (12 — 1)25=275 feet. The last chord of a spiral should have 
a smaller degree of curvature than the simple curve to which it 
is joined. If the curves are very sharp, such as are used in street 
work and even in suburban trolley work, an increase in degree 
of curvature of 1° per 25 feet will not be sufficiently rapM, a» 



§47. 



ALIGNMENT. 



49 



such a rate would require too long curves, 2°, 10°, or even 20° 
increase per 25 feet may be necessary, but then the chords 
should be reduced to 5 feet. Such 
a rapid rate of increase is justified 
by the necessary reduction in 
speed. On the other hand, very 
high speed will make a lower rate 
of increase desirable, and there- 
fore a spiral whose degree of cur- 
vature increases only 0° 30' per 25 
leet may be used. Such a spiral 
would require a length of 375 feet 
to run on to an 8° curve, which is 
inconveniently long, but it might 
be used to run on to a 4° curve, 
where its length would be only 175 
feet. Three spirals have been de- 
veloped in Table IV, each with 
chords of 25 feet, the rate of in- 
crease in the degree of curvature 
being 0° 30', 1° and 2° per chord. 
One of these will be suitable for 
any curvature found on ordinary 
steam-railroads. 

47. To find the ordinates of a 
i°-per-25-feet spiral. Since the 
first chord subtends a 1° curve, its central angle is 0° 15' and the 
angle aQV (Fig. 31) is 7' 30". The tangent at a makes an 
angle of 15' with VQ. The angle between the chord ba and the 
tangent at a is K30')=15', and the angle 6a6" = K30') + 15' 
= 30'. Similarly 

the angle c&c" = K45') +30' + 15' = 67' 30" = 1° 07' 30", 
and the angle dcd" =2° 0'. 

The ordinate aa'= 25 sin 7' 30", and 
Qa'=25 cos 7' 30". 
Q&'=Qa' + a'6' 

=25 (cos 7' 30" + cos 30'). 

=25 (sin 7' 30" + sin 30'), 
Similarly the ordinates of c, d, etc., may be obtained. 




Fig. 31. 



50 



EAILROAD CONSTRUCTION. 



§48. 



48. To find the deflections from any point of the spiral. 
aQF = 7'30''. Tan hQV=by-^Q¥; tan cQV^^cc' ^Qc'; etc. 
Thus we are enabled to find the deflection angles from the tan- 
gent at Q to any point of the spiral. 

The tangent to the curve at c (Fig. 32) makes slk angle of 
V 




Fig. 32. 
1°30' with QV, or cmV = 1°S0\ Qcm=cmV -cQm, The 
value of cQm is known from previous work. The deflection 
from c to Q then becomes known. 

acm = cmV—cap = cmV—caq—qap. caq is the deflection an- 
gle to c from the tangent at a and will have been previously 
computed numerically, gap = 15'. acm therefore becomes 
known. 

6cm = i of 45' =22' 30''; 
dcn = iof 60'=30'. 



§49. 



ALIGNMENT. 



51 



ecn=ec6!' —ncd" , ncd^^=^cmV, tan ecd" = {ee' —6!^d'^-^c'e\ all 
of which are known from the previous work. 

By this method the deflections from the tangent at any point 




Fig. 33. 

of the curve to any other point are determinable. These values 
are compiled in Table IV. The corresponding values of these 
angles when the increase in the degree of curvature per chord 
length is 30', and when it is 2°, are also given in Table IV. 

49. Connection of spiral with circular curve and with tangent. 
See Fig. 33.* Let AV and 5F be the tangents to be connected 



* The student should at once appreciate the fact of the necessary distor- 
tion of the figure. The distance MM' in Fig. 33 is perhaps 100 times its real 
proportional value. 



52 RAILROAD CONSTRUCTION. § 49. 

by a D° curve, having a suitable spiral at each end. If no 
spirals were to be used, the problem would be solved as in simple 
curves giving the curve AMB Introducing the spiral has the 
effect of throwing the curve away from the vertex a distance 
MM^ and reducing the central angle of the Z)° curve by 2(j). 
Continuing the curve beyond Z and Z^ to .4' and B\ we will 
have AA^ = BB^ = MM\ ZK=ihe x ordinate and is therefore 
known. Call MM^=m. A'N =x—R vers ^. Then 



m==MM'=AA' = r-T = TT-^ (33) 

cos JJ cos i J 



NA =AA^ sin \A={x — R vers 4) t-an J J. 

VQ=^QK-KN ^ NA-¥AV 

=y—R sin (j>A-{x—R vers (56) tan \d-\-R tan J J 

'=2/~-^ sin 0+a: tan Ji + i? cos ^ tan ii (34) 



When A'N has already been computed, it may be more con- 
venient to write 



yQ=2/ + jf^(taniJ-sin^)+AWtanJJ (35) 

' w , ^ ^ vers ^ .«^v 

=i?exseciiH j-r j-p^. ..... (36) 

cos J J cos JJ 



AQ«yQ-AF 

=2/ — -^ s^n ^+(a: — i^ vers ^) tan JJ (37) 

A method of obtaining the necessary dimensions using tables, 
is given in § 53a. 

Example. To join two tangents making an angle of 34° 20' 
by a 5° 40' curve and suitable spirals. Use l°-per-25-feet spirals 
with five chords. Then <jS = 3° 45', a; = 2.999, iJ-17° 10', and 
2/ -=124.942. 



§50.' 
[Eq. 33] 



[Eq. 36] 



[Eq. 35] 



ALIGNMENT. 




53 


2.166 

x= 2 999 

A'iV= 0.833 


R 

vers 

cos J J 


3.00497 
7.33063 
0.33560 

9.92064 
9.98021 


m^MM'=AA' - 0.872 


R 
3xsec J J 


9.94043 


( 


3.00497 
8.66863 


FM=47.164 

m= 0.872 

yM' = 48.036 




1.67360 


y = 124 . 942 nat . tan J J = 
nat. sin ^ = 


=.30891 
=.06540 
.24351 
R 


9.38651 
3.00497 


246.314 


tan ji 

taii §i 
AV 


2.39148 


[See above] 
0.257 


9.92064 
9.48984 
9.41048 


FQ=371.513 - 
312 471 


^.0049? 
9.4S984 
2.49481 



[Eq. 37] 



AQ^ 59.042 

5di Fieild-v^dfk* When the' spiral is designexl diifirig the 
original Idcaticin^ th6 i^rigdrit dist^nee VQ shbuld B^ domputed 
and the p6int Q lad^ted. It is hardl}^ necessary tb Ideate all of 
the points df the spiral uhtil the track is io be laid: The e^- 
ttemities shduld be Ideated, and as there will usually be dne 
Eind perhaps two full station points on the spiral, these should 
also b6 located. Z may be located by setting dff QK^y arid 
KZ^Xj of else by the tabular deflection for Z from Q and the 
distance ZQy which is the long chord. Setting up the instrument 
at Z and sighting back at Q with the proper deflection, the tan- 
gent at Z may be found and the circular curve located as usual, 
its central angle being J— 20. A similar operation will locate 
Q' from Z\ 

To locate points on the spiral. Set up at Q, with the plates 



54 



RAILROAD CONSTRUCTION. 



50. 



reading 0° when the telescope sights along VQ. Set off from 
Q the deflections given in Table IV for the instrument at Q, 
using a chord length of 25 feet, the process being like the method 
for simple curves except that the deflections are irregular. If 
a full station-point occurs within the spiral, interpolate between 
the deflections for the adjacent spiral-points. For example, 
a spiral begins at Sta. 56 + 15. Sta. 57 comes 10 feet beyond 
the third spiral point. The deflection for the third point is 
35' 0''; for the fourth it is 56' 15". if of the difference 
(21' 15") is 8' 30"; the deflection for Sta. 57 is therefore 43' 30". 
This method is not theoretically accurate, but the error is small. 
Arriving at Z, the forward alignment may be obtained by sight- 
ing back at Q (or at any other point) with the given deflection 




, Fig. 34. 



for that point from the station occupied. Then when the plates 
read 0° the telescope will be tangent to the spiral and to the 
succeeding curve. All rear points should be cheeked from Z, 
If it is necessary to occupy an intermediate station, use the de- 
flections given for that station^ prienting as just explained for Z, 



§ 51. ALIGNMENT. 55 

checking the back points and locating all forward points up to Z 
if possible. 

After the center curve has been located and Z' is reached, the 
other spiral must be located but in reverse order, i.e., the sharp 
curvature of the spiral is at Z' and the curvature decreases 
toward Q\ 

51. To replace a simple curve by a curve with spirals. This 
may be done by the method of § 49, but it involves shifting the 
whole track a distance m, which in the given example equals 
0.87 foot. Besides this the track is appreciably shortened, 
which would require rail-cutting, But the track may be kept 
at practically the same length and the lateral deviation from the 
old track may be made very small by slightly sharpening the 
curvature of the old track, moving the new curve so that it is 
wholly or partially outside of the old curve, the remainder of it 
with the spirals being inside of the old curve. It is found by 
experience that a decrease in radius of from 1% to 5% will 
answer the purpose. The larger the central angle the less the 
change. The solution is as indicated in Fig. 34. 

0'iV=i^' cos ^-fx. 
0'7=0'A^sec^J 

= 72' cos ^ sec \d^x sec J J. 
m^MM'=MV-M'V 

-72exseciJ-(0'y-EO 

:^i2exsec ii— i^' cos ^sec Ji— :r sec Ji+iB'. . . . (38) 
AQ = QK-KN-\-NV-VA 

=y—R^ sin (j>-\-(R' cos ^ + 0^) tan \A—R tan \A 

=-2/--R' sin 9S-l-E'cos ^tan JJ-(i2-a:) tan ^i. . . (39) 

J 
The length of the old curve from Q to Q'=2^Q + 100-. 

The length of the new curve from Q to Q'=2L + 100 ~r , 

in which L is the length of each spiraL 

Example, Suppose the old curve is a 7° 30' curve with a 
central angle of 38° 40'. As a trial, compute the relative length 
of a new 8° curve with spirals of seven chords, ^=7°0'; 
iJ = 19°20'; R (for the 7° 30' curve) =764.489; R' (for the 
8° curve) =716.779; a;=7,628. 



56 RAILROAD CONSTRUCTION. § 52. 

[Eq. 38] R 2.88337 

exsec ^ J 8.77642 

45.687 1.65979 

i2^ = 716.779 ^ ■ 

762.466 R' 2.85538 

cos <^ 9 . 99675 

sec i J 0.02521 

753.953 2 . 87734 

a; 0.88241 

sec U 0- 02521 

8.084 . 90762 

762.037 762.037 

m'^ 0.429 

[Eq.39] 27=174.722 R' 2.85538 

sin^ 9 . 08589 

87.353 1.94128 

R' 2.85538 

cos ^ 9 . 99675 
tan ii 9.54512 

249.606 2.39725 

22 = 764.489 

x^ 7.628 

Wf till 

265.543 . 2.42413 

424.328 352.896 
352.896 
^Q = 71.432 

The length of the old curve from Q to Q' is 

100-i=»100?^ « 515.556 

2^Q = 2 X 71.432 « ....... 142.864 

658.420 
New curve: 100-^ = 100 38.667^-^14.000 ^ 3^^ 333 

2L = 2 X 175 = 350.000 

658.333 658.333 
Difference in length = . 087 

Considering that this difference may be divided among 22 
joints (using 30-foot rails) no rail-cutting would be necessary. 
If the difference is too large, a slight variation in the value of 
the new radius R' will reduce the difference as much as neces- 
sary. A truer comparison of the lengths would be found by 
comparing the lengths of the arcs. 

52. Application of transition curves to compound curves. 
Since compound curves are only employed when the location is 
limited by local conditions, the elements of the compound curve 
should be determined (as in §§38 and 39) regardless of the 



§52. 



ALIGNMENT. 



57 



transition curves, depending on the fact that the lateral shifting 
of the curve when transition curves are introduced is very small. 
If the limitations are very close, an estimated allowance may be 
made for them. 

Methods have been devised for inserting transition curves 
between the branches of a compound curve, but the device is 




Fig. 35. 



complicated and usually needless, since when the train is once 
on a curve the wheels press against the outer rail steadily and* 
a change in curvature will not produce a serious jar even though 
the superelevation is temporarily a little more or less than it 
should be. 



58 RAILROAD CONSTRUCTION. § 53. 

If the easier curve of the compound curve is less than 3° or 
4°, there may be no need for a transition curve off from that 
branch. This problem then has two cases according as transition 
curves are used at both ends or at one end only. 

a. With transition curves at both ends. Adopting the method 
of § 49, calKng J^^^J, we may compute mi=Milf/. Similarly, 
calling A 2 = ^ A J we may compute m2=MM2, But iW/ and M^ 
must be made to coincide. This may be done by moving the 
curve Z'My' and its transition curve parallel to Q'V a distance 
M/M3, and the other curve parallel to QF a distance Mj'ikfg. 
In the triangle ikf/MgMg', the angle at ikf/=90° — J„ the angle 
at ikf2'=90° — ^2, and the angle at M^ = A. 



rr^i Ti^/T.^ T.^/71^/ sin (90° — io) / N cos A^ 

Then M^'M^=M^'M2' ^^% — -. — ^^ ={m^-m2)-r ^ 



sin A V 1 2 g- j^ J • 

Similarly Mn'M. ^M^M^ ^^ — - — ^ = (m^—m^j—. — y. 

•^2 3 12 J \ 1 2/ J 



K40) 



b. With a transition curve on the sharper curve only. Com- 
pute mi=MM/ as before; then move the curve Zj^Myf parallel 
to Q^V a distance of 

M/M,=mi-?^^ . (41) 

sm A •^ 

The simple curve MA is moved parallel to VA a distance of 

MM.^m,''-^. . ■ (42) 

* ^ sm J 

If Ji and ^2 are both small, Tlf /ikf^ and MAf^ may be more 
than m,, but the lateral deviation of the new curve from the old 
will always be less than m^, 

53. To replace a compound curve by a curve with spirals. 
The numerical illustration given below employs another method. 
We first solve for m^ for the sharper branch of the curve, plac- 
ing Ai = y in Eq. 38. A value for i?/ ^ay be found whose 
corresponding value of m^ will equal m^. Solving Eq. 38 for i2', 
we obtain 



P^ . Rvers ^A — m cos^A^x ^^^v 

"" cos ^ — cos J J 



§ 53. ALIGNMENT. 59 

Substituting in this equation the known value of rui {=m^ 
and calHng R' = R2, R=R2, and A2 = ^A, solve for T^g'- Obtain 
the value of AQ for each branch of the curve separately by Eq. 
39, and compare the lengths of the old and new lines. 

Example, Assume a compound curve with Z)i=8°, Z)2=4°, 
ii=36°, and ^2 = 32°. Use l°-per-25-feet spirals; ^i = 7°0'; 
^2 = 1° 30'. Assume that the sharper curve is sharpened from 
8° 0' to 8^ 12'. 

[Eq.38] Ri 2.85538 

exsec 36° 9.37303 

169.209 2.22842 

i?i' = 699.326 ^ ^ , 

sec a\ 0-09204 
857.970 2.93347 



0.88241 
. 09204 
9.429 " . 0.97445 



sec Ji 0.09204 



867.399 867.399 
mi= 1.136 



[Eq.43] R2 3.15615 

vers 32° 9.18170 

217.700 2 . 33785 

mi = l 136 0.05538 

^^ cos 32° 9-92842 

0.963 9.98380 

a;2= 0.763 

1.726 1.726 .==• 

215.974 2.33440 

nat. cos = . 99966 
nat. cos ^2= .84805 

.15161 9.18073 

i?2' = 1424.54 [4°1'22"] 3.15367 

[Eq.39] .. = 174.722 ^, ^=^= 

sin ^1 9.08589 

85.226 1.93057 

2?i' 2.84468 

cos^i 9.99675 

tan ^A [ii = 36°] 9.86126 

604.302 2.70269 

i2i = 716.779 
a:i= 7.628 

709.151 2.85074 

6797024 '^^^' ^-^5126 

600.461 515.235 2.71200 

AQi- 78.563 600.461 



60 RAILROAD CONSTRUCTION. § 53. 

[Eq. 39] K2' 3.15367 

2/2= 74.994 sin ^2 8.41792 

37.290 1.57159 

R/ 3.15367 

cos <f>2 9 . 99y85 

tan iJ(J^ = 32°) 9.79579 

889.843, 2.9-^931 

2^2 = 1432.69 
X2= 0.76 

1431.93 3.15592 

tan ^J 9.79579 

894 . 770 o . . . 2.95171 

964.837 932.060 
932.060 

AQ2= 32.777 
For the length of the old track we have : 

100^' = 100 ^J= 450. 
Ui o 

100-^^ = 100^^^= 800. 

^Qi = 78 563 
AQ2 = 32.777 
1361.340 
For the length of the new track we have ; 

100-^^' = 100-g||g-= 353.659 

100^^=10of^= 758.140 

Spi ral on 8° 1 2' curv'e 1 75 . 000 
*' " 4° 01' 22'' " 75. 

Length of new track = 1361,799 

" " old " == 1361.340 

Excess in l6ngth of new track = 0.459 feet. 

Since the new track is slightly longer than the old, it showB 
that the new track runs too far outside the old track at the 
P££. On the other hand the offset m is only 1.136. Th^ 
maximum amount by which the new track comes inside of the 
'old track at two points, presumably not far from Z' and Z, is 
very difficult to determine exactly. Since it is desirable that 
the maximum offsets (inside and outside) should be made as 
nearly equal as possible, this feature should not be sacrificed to 
an effort to make the two lines of precisely equal length so that 
the rails need not be cut. Therefore, if it is found that the offsets 
inside the old track are nearly equal to m (1.136), the above 
figures should stand. Otherwise m ma}^ be diminished (and the 
above excess in length of track diminished) by increasing R^' 
very slightly and making the necessary consequent changes. 



§ 53a. ALIGNMENT. 61 

53a. Use of Table IV. Prof. R. B. H. Begg, of Syracuse 
University, has submitted to the author a series of tables 
which will materially simplify the work of solving Equations 
(33) to (37), and which have been added to Table IV of pre- 
vious editions. Since these equations involve R and J (which 
may each have any values) in combination with several values 
of ^, it would require impracticably extensive tables to give 
precise values of the required dimensions for any possible com- 
bination of Rf Jj and (j). But the tables may be utilized hy 
interpolation with all necessary accuracy within their range. 
Rules for the use of the tables and for the field work are as 
follows: 

1. Find P. C. (point A) as if no transition curve were to be 

used. 

2. Lay off the distance AN (part C of table) to iV; then offset 

the distance A'N (part B) to A\ the new P. C; from A^ 
measure a distance NQ (part B) to Q. 

3. Set transit on A'; sight parallel to tangent and run in 

circular curve, setting Z from deflection and distance 
(part S); or -^ can be set by measuring ZK and QK 
(part B) from Q, 

4. Set transit on Q, sight along tangent and turn the deflection 

(part A) for each 25-foot station^ for as many chord 
lengths as required; or the points may be located by 
measuring distances ij 'along the tangent from Q and off- 
setting the ctrrresponding distances :3c 

VEKtlCAL dimvESi 

54. Necessity for their tis'e* Whenever thefe is a cMnge in 
the rate of grade, it is necessary to eliminate the angle that 
Would be formed at the point of change and to connect the two 
grades by a curve. This is especially necessarv^ at a sag between 
two grades, since the shock caused by abruptly forcing an up- 
ward motion to a rapidly moving heavy train is very severe both 
to the track and to the rolling stock. The necessity for vertical 
curves was even greater in the days when link couplers were in 
universal use and the ^' slack '^ in a long train was very great. 
Under such circumstances, when a train was moving down a 
heavy grade the cars would crowd ahead against the engine. 
Reaching the sag, the engine Would begin to pull out, rapidly 
taking out the slack. Six inches of slack on each car would 
amount to several feet on a long train, and the resulting jerk on 



62 RAILROAD CONSTRUCTION. § 54. 

the couplers, especially those near the rear of the train, has fre- 
quently resulted in broken couplers or even derailments. A 
vertical curve will practically eliminate this danger if the curve 
is made long enough, but the rapidly increasing adoption of 
close spring couplers and air-brakes, even for freight trains, is 
obviating the necessity for such very long curves. 

55. Required length. Theoretically the length should de- 
pend on the change in the rate of grade and on the length of the 
longest train dii the road. A sharp change in the rate of grade 
requires a long curve; a long train requires a long curve; but 
since the longest trains are found on roads with light grades and 
small changes of grade, the required length is thus somewhat 
equalized. The A.R.E.A. rule is: '^On class A roads (see § 198) 
rates of change of 0.1 per cent per station on summits and 0.05 per 
cent per station in sags should not be exceeded. On minor roads 
0.2 per cent per station on summits and 0.1 per cent per station 
in sags may be used.'^ When changing from a down grade to an 
up grade (or vice versa) the change of grade equals the numerical 
sum of the two rates of grade. For example, if a 0.5 per cent 
down grade is followed by a 0.7 per cent up grade, the road being 
a "minor" road, then, by the above rule the length of the curve 
should be at least [0.5 -(-0.7)] -^0.1 =12 stations or 1200 feet. 
Added length increases the amount of earthwork required both 
in cuts and fills, but the resulting saving in operating expenses 
will always justify a considerable increase. 
56. Form of curve. In Fig. 36 assume that A and C, equi- 



distant from B, are the extremities of the vertical curve. Bisect 
AC Sit e; draw Be and bisect it at h. Bisect AB and BC at k 
and L The line kl , will pass through h. A parabola may be 
drawn with its vertex at h which will be tangent to AB and BC 
at A and C. It may readily be shown * from the properties of 
a parabola that if an ordinate be drawn at any point (as at n) 
we will have 

* See noteaOoot of p. 64, 



li 



§ 57. ALIGNMENT. 63 

sn : eh (or KB) : : Ant^ : Ae^ 
or sn = eh . . . o . . . f44> 

In Fig. 36 the grades are necessarily exaggerated enormously. 
With the proportions found in practice we may assume that 
ordinates (such as mf, eB, etc.) are perpendicular to either 
grade, as may suit our convenience, without any appreciable 
error. In the numerical case given below, the variation of 
these ordinates from the vertical is 0° 07', while the effect of 
this variation on the calculations in this case (as in the most 
extreme cases) is absolutely inappreciable. It may easily be 
shown that the angle CAB=hali the algebraic difference of the 
rates of grade. Call the difference, expressed in per cent of 
grade, r; then CAB = ^r. Let Z=length (in "stations" of 100 
feet) of the line AC, which is practically equal to the horizontal 
measurement. Since the angle CAB is one-half the total change 
of grade at B, it follows that Be = ^lX ir Therefore 

Bn=ilr (44a) 

Since Bh ^or eh) are constant for any one curve, the correction 
sn at any point (see Eq. 44) equals a constant times Am'^. 

57. Numerical example. Assume that B is located at Sta. 
16+20; that the grade of AB is -0.5%, and of BC +0.7%; 
also that the elevation of B above the datum plane is 162.6. 
Then the algebraic difference of the grades, r, =0.7 — (—0.5) = 
1.2; Z = 12. m = iZr = JXl2Xl.2 = 1.8. A is at Sta. 10+20 
and its elevation is 162.6 + (6X0.5) =165.6; C is at Sta. 22+20 
and its elevation is 162.6 + (6X0.7) =166.8. The elevation of 
Sta. 11 is found by adding sn to the elevation of s on the 
straight ^grade Hne. The constant] (e/i^Ae ) equals in this case 
Therefore the curve elevations are 

A, Sta. 10 + 20, 162.6+(6.00X0.5) =165.60 

11 165.6-(0.80X0.5) + 2ooW 802=165.23 

12 165.6-(1.80X0.5) + oooW 1802 =164.86 

13 165 . 6 - (2 . 80 XO . 5) + 2 ooVoo 2802 = 164 . 59 

14 ' 165. 6 -(3. 80X0. 5) +2xyoWo 3802=164.42 

15 165.6 -(4.80X0.5) + 200W 4802 =164.35 

16 165.6-(5.80X0.5) + Wo^^ 5802 =164.38 



64 RAILROAD CONSTRUCTION. § 57* 

B. 16+20, 162.6+ 1.80 =164.40 

17 166.8 -(5.20X0.7) + Wu^ 5202 =164,51 

18 166.8 -(4.20X0.7) + 2ouWa 4202 =164.74 

19 166.8-(3.20X0.7) + Woon3202=165.07 

20 166 . 8 - (2 . 20 XO . 7) + ^^.^^^ 2202 = 165 . 50 

21 166.8-(1.20X0.7) + 5ffo^ou 1202=166.03 

22 166.8 -(0.20X0.7) + 2^^n 202=166.66 

C, 22+20, 162.6+(6. 00X0. 7) =166.80 



DEMONSTRATION OF EQ. 44. 

The general equation of a parabola passing through the point n (Fig. 36) 
may be written 

2/2 + 2/,j2 == 2p{x + x^)i 

from which ar^ =-^ — 1- --^ x. 

"'' 2p 2p 

When « =» ic^ y '^ Va ^^^ ^^ have 

^n^ 2p'^ 2p ^A. 

The general equation of a tangent passing through the point A may be 
written 

VVa " P(^ + ^^)' 



from which 








X = 


p ■^* 




When 25 == a;^, 


y 


"VsV 


= 2/ J 


and 


we have 

ynyA 

p -^A, 








sn ^ 


^n- 


2p = 

871 = 


yj + yn^- 

2p 

{yA-y^y 

2p 
yA' Xe^ 
^A eh 

-J Am 
eh - . 

.4e^ 


2p' 



This proves the general proposition that if secants are drawn parallel to 
the axis of x, intersecting a parabola and a tangent to it, the intercepts be- 
tween the tangent and the parabola are proportional to the square of the 
distances (measured parallel to y) from the tangent point. 



II 



CHAPTER III. 

EARTHWORK. 
FORM OF EXCAVATIONS AND EMBANKMENTS. 

58. Usual form of cross-section in cut or fill. The normal 
form of cross-section in cut is as sho^sm in Fig. 37, in which 
e . . .g represents the natural surface of the ground, no matter 



e \ 




how irregular; ah represents the position and w^idth of the re« 
quired roadbed; ac and bd represent the ^'side slopes'' which 
begin at a and b and which intersect the natural surface at such 




Fig. 38. 



d 



points {c and 6^ as will be determined by the required slope 
angle (/?). 

The normal section in fill is as shown in Fig. 38. The points 
c and d are likewise determined by the intersection of the re- 

65 



66 



RAILROAD CONSTRUCTION. 



59. 



quired side slopes with the natural surface. In ease the required 
roadbed (ah in Fig. 39) intersects the natural surface, both cut 




Fig. 39. 

and fill are required, and the points c and d are determined as 
before. Note that /? and ^' are not necessarily equal. Their 
proper values will be discussed later. 

59. Terminal pyramids and wedges. Fig. 40 illustrates the 
general form of cross-sections when there is a transition from 
cut to fill, a.^.g represents the grade line of the road which 




Fig. 40. 

*3asses from cut to fill at d. sdt represents the surface profile. 
A cross-section taken at the point where either side of the road- 
bed first cuts the surface (the point m in this case) will usually 
be triangular if the ground is regular. A similar cross-section 
should be taken at o, whore the other side of the roadbed cuts 
the surface. In general the earthwork of cut and fill terminates 



§ 60. EARTHWOHK. 67 

in two pyramids. In Fig. 40 the pyramid vertices are at n 
and k, and the bases are Ihm and opq. The roadbed is generally 
wider in cut than in fill, and therefore the section Ihm and the 
altitude In are generally greater than the section opq and the 
altitude pk. When the line of intersection of the roadbed and 
natural surface (nodkm) becomes perpendicular to the axis of 
the roadbed (ag) the pyramids become wedges whose bases are 
the nearest convenient cross-sections. 

6o. Slopes, a. Cuttings. The required slopes for cuttings 
vary from perpendicular cuts, which may be used in hard rock 
which will not disintegrate by exposure, to a slope of perhaps 
4 horizontal to 1 vertical in a soft material like quicksand or in 
a clayey soil which flows easily when saturated. For earthy 
materials a slope of 1 : 1 is the maxim.um allowable, and even 
this should only be used for firm material not easily affected by 
saturation. A slope of IJ horizontal to 1 vertical is a safer 
slope for average earthwork. It is a frequent blunder that 
slopes in cuts are made too steep, and it results in excessive work 
in clearing out from the ditches the material that slides down, 
at a much higher cost per yard than it would have cost to take 
it out at first, to say nothing of the danger of accidents from 
possible landslides. 

b. Embankments. The slopes of an embankment vary from 
1 : 1 to 1.5 : 1. A rock fill will stand at 1 : 1, and if some care 
is taken to form the larger pieces on the outside into a rough 
dry wall, a much steeper slope can be allowed. This method is 
sometimes a necessity in steep side-hill work. Earthwork em- 
bankments generally require a slope of 1^ to 1. If made 
steeper at first, it generally results in the edges giving way, re- 
quiring repairs until the ultimate slope is nearly or quite IJ : 1. 
The diffi-culty of incorporating the added material with the old 
embankment and preventing its sliding off frequently makes 
these repairs disproportionately costly. 

6i. Compound sec tioils. . When the cut consists partly of 
earth and partly of rock, a compound cross-section must be 
made. If borings have been made so that the contour of the 
rock surface is accurately known, then the true cross-section may 
be determined. The rock and earth should be calculated sepa- 
rately, and this will require an accurate knowledge of where the 
rock ''runs out'' — a difficult matter when it must be deter- 



68 RAILROAD CONSTRUCTION. § 62. 

mined by boring. During construction the center part of the 
earth cut would be taken out first and the cut widened until a 
sufficient width of rock surface had been exposed so that the 
rock cut would have its proper width and side slopes. Then the 
earth slopes could be cut down at the proper angle. A^'berm" 
of about three feet is usually left on the edges of the rock cut as 




Fig. 41. 

a margin of safety against a possible sliding of the earth slopes. 
After the work is done, the amount of excavation that has been 
made is readily computable, but accurate preliminary estimates 
are difficult. The area of the cross-section of earth in the figUrie 
must be determined by a method similar to that developed for^ 
borrow-pits (see § 89). 

62. Width Off foadbed. Owing to the large and oftert dis- 
proportionate addition to volume of cut or fill caused by the 
addition of even one foot to the width of foadbed, there is a 
natural tendency to deduce the width until embankments bec^ome 
Unsafe and c^uts afe too narrow foi* proper drainage. The cdst 
of maintenance of roadbed is so largely dependent bn thd drain- 
age of the roadbed that there is tni6 economy in making an 
ample allowance for it. The practice of some of the leading 
railroads of the country in this respect is given in the following; 
table, in which are also given some data belonging more properly 
to the subject of superstructure. 

It may be noted from the table that the average width 
for an earthwork cut, single track, is about 24.7 feet, with a 
minimmn of 19 feet 2 inches. The widths of fills, single track, 
average over 18 feet, with numerous minimums of 16 feet; 
The widths for double track may be found by adding the distance 
between track centers, which is usually 13 feet. 



62. 



EARTHWORK. 



69 



g ^ O (1) 



lo io ^ »o »o «:> lo lo lo lo »o »o 






*iou:>ioioio " io '"lOJO^ ' 



• .4*Hn • 1— i 



o 

ft 






XXX 

+++ 



00O(M 

^ COCO 



% CO • 

ooXx 

^co^^ 
co + ^ 



<N 



c3 



CO T^ 05 ^ 

CO CO (M tH 



CO COXOCco 

Q g 



?3 



d cXXX.^^,<^ 

« S(N^(N^.SoX 



bias' 









<1 OqqSWh^^h^;^!z; 









o 



CO d 

fl O 



70 RAILROAD CONSTRUCTIONo § 64. 

63. Form of subgrade. Specifications (or the cross-section 
drawings) formerly required that the subgrade should have a 
curved form, convex upward, or that it should slope outward 
from a slight ridge in the center, with the evident purpose of 
draining to the sides all water which might percolate through the 
ballast. If the subsoil were hard and impenetrable by the ballast, 
the method might answer, but experience has shown that, with 
ordinary subsoils, the ballast immediately under each rail is 
forced a little deeper into the subsoil by the passage of each train. 
Periodical retamping of ballast under the ends of the ties, and 
little or no tamping under the center, only adds to the accumula- 
tion under each rail. A cross-section of a very old roadbed will 
frequently show twice as much depth of ballast under the rails 
as there is under the center. This method of tamping quickly 
obliterates the original line of demarcation between ballast and 
subsoil and any expected improvement in drainage due to sloping 
subsoil is not reahzed. Therefore the A.R.E.A. specifications 
call for flat subgrades. 

64. Ditches. *' The stability of the track depends upon the 
strength and permanence of the roadbed and structures upon 
which it rests; w^hatever will protect them from damage or pre- 
vent premature decay should be carefully observed. The worst 
enemy is water, and the further it can be kept away from the 
track, or the sooner it can be diverted from it, the better the 
track will be protected. Cold is damaging only by reason of 
the water which it freezes; therefore the first and most impor- 
tant provision for good track is drainage." (Rules of the Road 
Department, Illinois Central R. R.) 

The form of ditch generally prescribed has a flat bottom 12" 
to 24" wide and with sides having a minimum slope, except in 
rock-work, of 1 : 1, more generally 1.5 : 1 and sometimes 2:1. 
Sometimes the ditches are made V-shaped, which is objection- 
able unless the slopes are low The best form is evidently that 
which will cause the greatest flow for a given slope, and this 
. will evidently be the form in which the 
ratio of area to wetted perimeter is the 
largest. The semicircle fulfills this con- 
dition better than any other form, but the 
nearly vertical sides would be difficult to 
maintain. (See Fig. 42.) A ditch, with a flat bottom and such 




§ 65. EARTHWORK. 71 

slopes as the soil requires, which approximates to the circular 
forui will therefore be the best. 

When the flow will probably be large and at times rapid it 
will be advisable to pave the ditches with stone, especially if the 
soil is easily washed away. Six-inch tile drains, placed 2' under 
the ditches, are prescribed on some roads. (See Fig. 43.) No 
better method could be devised to insure a dry subsoil. The 
ditches through cuts should be led off at the end of the cut so 
that the adjacent embankment will not be injured. 

Wherever there is danger that the drainage from the land 
above a cut will drain down into the cut, a ditch should be made 
near the edge of the cut to intercept this drainage, and this 
ditch should be continued, and paved if necessary, to a point 
where the outflow will be harmless. Neglect of these simple 
and inexpensive precautions frequently causes the soil to be 
loosened on the shoulders of the slopes during the progress of a 
heavy rain, and results in a landslide which will cost more to 
repair than the ditches which would have prevented it for all 
time. 

Ditches should be formed along the bases of embankments; 
they facilitate the drainage of water from the embankment, 
and may prevent a costly slip and disintegration of the em- 
bankment. ^ 

65. Effect of sodding the slopes, etc. Engineers are unani- 
mously in favor of rounding off the shoulders and toes of em- 
bankments and slopes, sodding the slopes, paving the ditches, 
and providing tile drains for subsurface drainage, all to be put 
in during original construction. (See Fig. 43.) Some of the 
highest grade specifications call for the removal of the top layer 
of vegetable soil from cuts and from under proposed fills to 
some convenient place, from which it may be afterwards spread 
on the slopes, thus facilitating the formation of sod from grass- 
seed. But while engineers favor these measures and their 
economic value may be readily demonstrated, it is generally 
impossible to obtain the authorization of such specifications 
from railroad directors and promoters. The addition to the 
original cost of the roadbed is considerable, but is by no means 
as great as the capitalized value of the extra cost of mainte- 
nance resulting from the usual practice. Fig. 43 is a copy of 



72 



RAILROAD CONSTRUCTION. 



§65. 



designs * presented at a convention of the Ameiican Society of 
Civil Engineers by Mr. D. J. Whittemore, Past President of 
the Society and Chief Engineer of the Chi., Mil. & St. Paul 




PROPOSED SECTION OF ROADBED ON EMBANKMENT* 

GRAVEL^ 




Fig. 43. — *' Whittemore on Railway Excavation and Embankments " 
Trans. Am. Soc. C. E., Sept. 1894. 

R. R. The '^customary sections " represent what is, with some 
variations of detail, the practice of many railroads. The ^^ pro- 



* Trans. Am. Soc. Civil Eng., Sept. 1894. 



§ 66. EARTHWORK, 73 

posed sections'^ elicited unanimous approval. They should be 
adopted when not prohibited by financial considerations. 



EARTHWORK SURVEYS. 

66. Relation of actual volume to the numerical result. It 

should be realized at the outset that the accuracy of the result 
of computations of the volume of any given mass of earthwork 
has but little relation to the accuracy of the mere numerical 
work. The process of obtaining the volume consists of two 
distinct parts. In the first place it is assumed that the volume 
of the earthwork may be represented b}^ a more or less com- 
plicated geometrical form, and then, secondly, the volume of 
such a geometrical form is computed. A desire for simplicity 
(or a frank willingness to accept approximate results) will often 
cause the cross-section men to assume that the volume may be 
represented by a very simple geometrical form which is really 
only a very rough approximation to the true volume. In such 
a case, it is only a waste of time to compute the volume with 
minute numerical accuracy. One of the first lessons to be 
learned is that economy of time and effort requires that the 
accuracy of the numerical work should be kept proportional to 
the accuracy of the cross-sectioning work, and also that the 
accuracy of both should be proportional to the use to be made 
of the results. The subject is discussed further in § 94. 

67. Prismoids. To compute the volume of earthwork, it is 
necessary to assume that it has some geometric form whose vol- 
ume is readily determinable. The general method is to consider 
the volume as consisting of a series of prismoids, which are 
solids having parallel plane ends and bounded b}^ surfaces which 
may be formed by lines moving continuously along the edges of 
the bases These surfaces may also be considered as the sur- 
faces generated by lines moving along the edges joining the cor- 
responding points of the bases, these edges being the directrices, 
and the lines being always parallel to either base, which is a 
plane director. The surfaces thus developed may or ma}^ not 
be planes. The volume of such a prismoid is readily determin- 
able (as explained in § 70 et seq.), while its definition is so very 
general that it may be applied to very rough ground. The 
*Hwo plane ends" are sections perpendicular to the axis of the 
road. The roadbed and side slopes (also plane) form three of 



74 



RAILROAD CONSTRUCTION. 



68. 



the side surfaces. The only approximation lies in the degree of 
accuracy with which the plane (or warped) surfaces coincide with 
the actual surface of the ground between these two sections. 
This accuracy will depend (a) on the number of points which 
are taken in each cross-section and the accuracy with which the 
lines joining these points coincide with the actual cross-sections; 
(5) on the skill shown in selecting places for the cross-sections so 
that the warped surfaces shall coincide as nearly as possible with 
the surface of the ground. In fairly smooth country, cross- 
sections every 100 feet, placed at the even stations, are suf- 
ficiently accurate, and such a method simplifies the computations 
greatly; but in rough countr}^ cross-sections must be inter- 
polated as the surface demands. As will be explained later, 
carelessness or lack of judgment in cross-sectioning will introduce 
errors of such magnitude that all refinements in the computa- 
tions are utterly wasted. 

68. Cross-sectioning. The process of cross-sectioning con- 
sists in determining at an}^ place the intersection by a vertical 
plane of the prism of earth lying between the roadbed, the side 
slopes, and the natural surface. The intersection with the road- 




FiG. 44. 



bed and side slopes gives three straight lines. The intersection 
with the natural surface is in general an irregular line. On^ 
smooth regular ground or when approximate results are accept- 1 
able this line is assumed to be straight. According to the irreg- 



§ 69. EARTHWORK. 75 

ularity of the ground and the accuracy desired more and more 
''intermediate points'' are taken. 

The distance (d in Fig. 44) of the roadbed below (or above) 
the natural surface at the center is known or determined from 
the profile or by the computed establishment of the grade line. 
The distances out from the center of all " breaks " are deter- 
mined with a tape. To determine the elevations for a cut, set» 
up a level at any convenient point so that the line of sight is 
higher than any point of the cross-section, and take a rod read- 
ing on the center point. This rod reading added to d gives the 
height of the instrument (H. I.) above the roadbed. Sub- 
tracting from H. I. the rod reading at any ''break" gives the 
height of that point above the roadbed (hi, ki, hr, etc.). This 
is true for all cases in excavation. For fill, the rod reading at 
center minus d equals the H. L, which may be positive or nega- 
tive. When negative, add to the "H. I." the rod readings of 
the intermediate points to get their depths below ''grade"; 
when positive, subtract the "H. I." from the rod readings. 

The heights or depths of these intermediate points above or 
below grade need only be taken to the nearest tenth of a foot, 
and the distances out from the center will frequently be suffi- 
ciently exact when taken to the nearest foot. The roughness of 
the surface of farming land or^woodland generally renders use- 
less any attempt to compute the volume with any greater accu- 
racy than these figures would imply unless the form of the ridges 
and hollows is especially well defined. The position of the slope- 
stake points is considered in the next section. Additional dis- 
cussion regarding cross-sectioning is found in § 82. 

69. Position of slope-stakes. The slope-stakes are set at the 
intersection of the required side slopes with the natural surface,^ 
which depends on the center cut or fill (d). The distance of 
the slope-stake from the center for the lower side is x = ^b 
+ s(cZ+2/); for the up-hill side it is x^ = ^h-{-s(d—y^), s is the 
"slope ratio" for the side slopes, the ratio of horizontal to ver 
tical. In the above equation both x and y are unknown. There- 
fore some position must be found by trial which will satisfy the 
equation. As a preliminary, the value of x for the point a = hh 
•\-sd, which is the value of x for level cross-sections. In the 
case of fills on sloping ground the value of x on the down-hill 
side is grrea^er than this ; on the up-hUl side itis less. The differ- 
ence in distance is s times the difference of elevation. Take a 



76 KAILROAD CONSTRUCTION. § 69. 

numerical case corresponding with Fig. 45. The rod reading 
on c is 2.9; d=4.2; therefoie the telescope is 4.2— 2.9 = 1.3 
helow grade. s = 1.5 : 1, 6 = 16. Hence for the point a (or for 
level ground) a; = JX 16 + 1.5X4.2 = 14.3. At a distance out 
of 14.3 the ground is seen to be about 3 feet lower, which will 
not only require 1.5X3=4.5 more, but enough additional dis- 
tance so that the added distance shall be 1.5 times the additional 
drop. As a first trial the rod maj^ be held at 24 feet out and a 
reading of, say, 8.3 is obtained. 8.3 + 1.3=9.6, the depth of 
the point below grade. The point on the slope line (n) which 
has this depth below grade is at a distance from the center 





^—. 


1 1 




1 
1 


-p^^^ 


1 ^ / ^ 




XT- 


~/^ 7 \~~~^ 


■tttTTZ^^^^ 


5 


^ 1 y 


^^^^.^--^TT^T^T^TTT^T^^^ 


-% 


5^ 


V'n 







^^Ki 



Fig. 45. 

ic =8 + 1.5X9.6 =22.4. The point on the surface (s) having 
that depth is 24 feet out. Therefore the true point (m) is 
nearer the center. A second trial at 20.5 feet out gives a rod 
reading of, say, 7.1 or a depth of 8.4 below grade. This corre- 
sponds to a distance out of 20.6. Since the natural soil (espe- 
cially in farming lands or woods) is generally so rough that a 
difference of elevation of a tenth or so may be readily found hy 
slightly varying the location of the rod (even though the dis- 
tance from the center is the same), it is useless to attempt too 
much refinement, and so in a case like the above the combina- 
tion of 8.4 below grade and 20.6 out from center may be taken 
to indicate the proper position of the slope-stake. This is 
usually indicated in the form of a fraction, the distance out being 
the denominator and the height above (or below) grade being 
the numerator; the fact of cut or -fill may be indicated by C or jP. 
Ordinarily a second trial will be sufficient to determine with I 
sufficient accuracy the true position of the slope-stake. Ex- 
perienced men will frequently estimate the required distance! 



§ 70, EARTHWORK. 77 

out to within a few tenths at the first trial. The left-hand pages 
of the note-book should have the station number, surface eleva- 
tion, grade elevation, center cut or fill, and rate of grade. The 
right-hand pages should be divided in the center and show the 
distances out and heights above grade of all points, as is illus- 
trated in § 84. The notes should read up the page, so that when 
looking ahead along the line the figures are in their proper 
relative position. The '^ fractions'' farthest from the center 
line represent the slope-stake points. 

70. Setting slope-stakes by means of " automatic" slope-stake 
rods. The equipment consists of a specially graduated tape and 
a specially constructed rod. The tape may readily be prepared 
by marking on the back side of an ordinary 50-foot tape which is 
graduated to feet and tenths. Mark '^0" at ^'^b " from the tape- 
ring. Then graduate from the zero backward, at true scale, to 
the ring. Mark off '^feef and "tenths'' on a scale propor- 
tionate to the slope ratio. For example, with the usual slope 
ratio of 1.5:1 each ^' foot" would measure 18 inches and each 
''tenth'' in proportion. 

The rod, 10 feet long, is shod at each end and has an endless 
tape passing within the shoes at each end and over pullej^s — to 
reduce friction. The tape should be graduated in feet and 
tenths, from to 20 feet — the^ and 20 coinciding. By moving 
the tape so that is at the bottom of the rod — or (practically) 
so that the 1-foot mark on the tape is one foot above the bottom 
of the shoe, an index mark may be placed on the back of the 
rod (say at 15 — on the tape) and this readily indicates when the 
tape is "set at zero." 

The method of use may best be explained from the figure and 
from the explicit rules as stated. The proof is given for two 
assumed positions of the level. 

(1) Set up the level so that it is higher than the "center" 
and (if possible) higher than both slope-stakes, but not more 
than a rod-length higher. On very steep ground this may be 
impossible and each slope-stake must be set by separate positions 
of the level. 

(2) Set the rod-tape at zero (i.e., so that the 15-foot mark 
on the back is at the index mark). 

(3) Hold the rod at the center-stake (B) and note the read- 
ing (ni or n2). Consider n to be always plus; consider d to be 
plus for cut and minus for fill. 



78 



RAILROAD CONSTRtJCTION. 



§70. 



(4) Raise the tape on the face side of the rod (n + <^^Apphed 
literally (and algebraically), when the level is helow the roadbed 
(only possible for fill) , (n + d) = (ng + ( — <i/) ) = ^2 — (i/. This being 
numerically negative, the tape is lowered {df—n^. With level 
at (1), for fill, {n-\-d)= (ji^ + ( —df)) = (n^—df) ; this being positive, 
the tape is raised. With level at (1), for cut, the tape is raised 
(rii + dc). In every case the effect is the same as if the telescope 
were set at the elevation of the roadbed. 




Fig. 45a. 



(5) With the special distance-tape, so held that its zero is |& 
from the center, carry the rod out until the rod reading equals 
the reading indicated by the tape. Since in cut the tape is 
raised (n+c?), the zero of the rod-tape is always higher than the 
level (unless the rod is held at or below the elevation of the road- 
bed — which is only possible on side-hill work), and the reading 
at either slope-stake is necessarily negative. The reading for 
slope-stakes in fill is always positive. 

1 (6) Record the rod-tape reading as the numerator of a frac- 
tion and the actual distance out (read directly <from the other 
side of the distance-tape) as the denominator of the fraction. 

Proof. Fill. Level at (i). Tape is raised {n^—df). When 
rod is held at C/, the rod reading is +a:, which =rfi — {n^—d^. 
But the reading on the back side of the distance-tape is also x. 

Fill. Level at (2). Tape is raised (n2—df)j i.e., it is lowered 
(df—n^. When rod is held at C/, the rod reading is -\-Xj which 
similarly = rf2—(n2—df) = r/2 + {df—n^. Distance-tape as be- 
fore. 




§71. 



EARTHWORK. 



79 



Cut Level at (i). Tape is raised (rii+cZc). When rod is 
held at Cc the rod reading is— 2;, which = rci — (ni + 6?c), i.e., 
2 = (rii + dc) — Tci. The distance-tape will read z. 

Side-hill work. It is easily demonstrated that the method, 
when followed literally, may be applied to side-hill w^ork. al- 
though there is considerable chance for confusion and error, 
when, as is usual, \h and the slope ratio are different for cut and 
for fiU. 

The method appears complicated at first, but it becomes 
mechanical and a time-saver when thoroughly learned. The 
advantages are especially great when the ground is fairly level 
transversely, but decrease when the difference of elevation 
of the center and the slope-stake is more than the rod length. 
By setting the rod-tape ^ ' at zero," the rod may always be used 
as an ordinary level rod and the regular method adopted, as in 
§ 69. Many engineers who have thoroughly tested these rods 
are enthusiastic in their praise as a time-saver. 

COMPUTATION OF VOLUME. 



71. Prismoidal formula. Let Fig. 46 represent a triangular 
prismoid. The tw^o triangles forming the ends lie in parallel 
planes, but since the angles of one triangle are not equal to the 
corresponding angles of the other triangle, at least two of the 
surfaces must be warped. If a section, parallel to the bases, is 




Fig. 46. 



made at hnj point at a distance x from one end, the area of the 
section will evidently be 



80 RAILROAD CONSTRUCTION. § 71. 

The volume of a section of infinitesimal length will be Axdx, 
and the total volume of the prismoid will be * 

£Axdx^i£ Ui + (&2-&i)f h,-\- (h,-h,) jl dx 



{Ihh + i^A + V>2K + J&2^2] 



[i^l^l + i^iC^l + h) + i&2(^l + /^2) + i^2^2] 



z 

=-^Ai + 4A.;. + A2], (45) 

in which A^, A 2, and Am are the areas respectively of the two 
bases and of the middle section„ Note that Am is not the mean 
of Ai and A 2; although it does not necessarily differ very greatly 
from it. 

The above proof is absolutely independent of the values, ab- 
solute or relative, of h^j 62? ^1? oi" ^2- For example, I12 may be 
zero and the second base reduces to a line and the prismoid be- 
comes wedge-shaped; or 62 and /12 may both vanish, the second 
base becoming a, point and the prismoid reduces to a pyramid. 
Since every prismoid (as defined in § 67) may be reduced to a 
combination of triangular prismoids, wedges, and pyramids, and 



* Students unfamiliar with the Integral Calculus may take for granted the 

fundamental formulae that / dx = x, that / xdx = ^x'^, and that / x^dx = ^x^\ 

also that in integrating between the T.nits of I and (zero) , the value of the 
integral may be found by simply substituting I for x after integration. 



II 



§ 72. EARTHWORK. 81. 

since the formula is true for any one of them individually, it is 
true for all collectively ; therefore it may be stated that * 

The volume of a prismoid equals one sixth of the perpendicular 
distance between the bases multiplied by the sum of the areas of 
the two bases plus four times the area of the middle section. 

While it is always possible to compute the volume of any 
prismoid by the above method, it becomes an extremely comph- 
©ated and tedious operation to compute the true value of the 
middle section if the end sections are complicated in form. It 
therefore becomes a simpler operation to compute volumes by 
approximate formulae and apply, if necessary, a correction. 
The most common methods are as follows : 

72^ Averaging end areas. The volume of the triangular 
prismoid (Fig. 46), computed by averaging end areas, is 

Z 

— [ J&i/ii + i?>2^2]- Subtracting this from the true volume (as 

given in the equation above Eq. 45), we obtain the correction 
~[{b,-b,)(h,-h,)] (46) 

This shows that if either the h^s or b's are equal, the correc- 
tion vanishes; it also shows that if the bases are roughly similar 
and b varies roughly with h (w^hich usually occurs, as will be 
seen later), the correction wiil be negative, which means that the 
method of averaging end areas usually gives too large results. 

73. Middle areas. Sometimes the middle area is computed 
and the volume is assumed to be equal to the length times the 

middle area. This will equal — X ^ ^ X \ ^ Subtracting 

this from the true volume, we obtain the correction 

Y^\-^^){K-hi) (47) 

As before, the form of the correction shows that if either 
the K^ or 6's are equal, the correction vanishes; also under the 
usual conditions, as before, the correction is positive and only 
one-half as large as by averaging end areas. Ordinarily the 
labor involved in the above method is no less than that of 
applying the exact prismoidal formula. 

* The student should note that the derivation of equation (45) does not 
complete the proof, but that the statements in the following paragraph are 
logically necessary for a general proof. 



82 



BAILROAD CONSTRUCTION. 



§74. 



74. Two-level ground. When approximate computations of 
earthwork are sufficiently exact the field-work may be materi- 
ally reduced by observing simply the center cut (or fill) and the 
natural slope a, measured with a clinometer. The area of such 
a section (see Fig. 48) equals 

ab 



^(a + d)(xi + Xr) — 



2 • 



But 

from which 

Similarly, 
Substituting, 



Xi tsiii ^ = a + (!-{- xi tan a, 
a-^d 



Xr = 



Area = (a + dy 



tar 


l/?— tan a ' 




a + d 


tan 


/? + tan a ' 


^2 


tan/? 



ah 
~2 • 



(48) 



tan^ /?— tan^ o 

The values a, tan /?, tan^ /? are constant for all sections, so 
that it requires but little work to find the area of any section. 




Fig. 47. 

As this method of cross-sectioning implies considerable approxi- 
mation, it is generally a useless refinement to attempt to com- 




Fio. 48. 



^1 



§75. EARTHWORK. 83 

pute the voiume with any greater accuracy than that obtained 
by averaging end areas. It may be noted that it may be easily 
proved that the correction to be applied is of the same form as 
that found in § 72 and equals 

, ^[W + Xr')-W' -\-xn][{d'' -\-a)-{d^ -Va)l 

which reduces to 

When d" —dJ the correction vanishes. This shows that when 
the center heights are equal there is no correction — regardless 
of the slope. If the slope is uniform throughout, the form of the 
correction is simplified and is invariably negative. Under the 
usual conditions the correction is negative, i.e., the method 
generally gives too large results. 

75. Level sections. When the country is very level or when 
only approximate preliminary results are required, it is some- 
times assumed that the cross-sections are level. The method of 
level sections is capable of easy and rapid computation. The 
area may be written as 

{^a-^d)H-^ (50) 







Fig. 49. 
This also follows from Eq. 48 when a = and tan B=—, 

8 

8 here represents the " slope ratio," i.e., the ratio of the horizontal 
projection of the slope to the vertical. A table is very readily 
formed giving the area in square feet of a section of given depth 
and for any given width of roadbed and ratio of side-slopes. 



84 ' RAILROAD CONSTRUCTION. § 76. 

The area may also be readily determined (as illustrated in the 
following example) without the use of such a table; a table of 
squares will facilitate the work. Assuming the cross-sections 
at equal distances (=/) apart, the total approximate volume 
for any distance will be 

^[Ao-{-2{A,+A,-h ■. ^An-d+^nl . . . (51) 



The prismoidal correction may be directly derived from 

12^ 



Eq. 46 as J^[2(a + d')s-2{a + d'')s][(a + d'')-(a + d')l which 



reduces to 



_|(^'_^/n2 or -— l(c^'_d'0^ . . (52) 



This may also be derived from Eq. 49, since a = 0, tan a=0, 
and tan/?=2a-^6 This correction is always negative, showing 
that the method of averaging end areas, when the sections ar^ 
level, always gives too large results. The prismoidal correction 
for any one prismoid is therefore a constant times the square 
of a difference. The squares are always positive whether the 
differences are positive or negative. The correction therefore 
becomes 

-^^Kd'^d'T. (53) 



76. Numerical example: level sections. Given the following 
center heights for the same number of consecutive stations lOG 
feet apart; width of roadbed 18 feet; slope IJ to 1. 

The products in the fifth column may be obtained very 
readily and with sufficient accuracy by the use of the slide-rule 
described in § 79. The products should be considered as 

(a-^d)(a + d)-i--. In this problem s = li, — = .6667. To apply 

the rule to the first case above, place 6667 on scale B over 89 
on scale Aj then opposite 89 on scale B will be found 118.8 on 
scale A, The position of the decimal point will be evident from 
an approximate mental solution of the problem. 



§77. 



EAETHWORK. 



Sta. 


Center 
Height. 


a + d 


(a-\-dy 


(a + dys 


Areas. 


d'~^'' 


(d'^d'')^ 


17 


2.9 


8.9 


79.21 


118.81 


118 


81 






18 


4.7 


10.7 


114.49 


171.74 1 


f343 


48 


1.8 


3.24 


19 


6.8 


12.8 


163.84 


245.76 ! 
469.93 r 


y2-^^9^ 

^^^ 1 939 


52 


2.1 


4.41 


20 


11.7 


17.7 


313.29 


86 


4.9 


24.01 


21 


4.2 


10.2 


104.04 


1 56 . 06 J 


1312 


12 


7.5 


56.25 


22 


1.6 


7.6 


57.76 


86.64 


86 


64 


2.6 


6.76 



ab 
2 ' 



6X18 



= 54 
1752.43X100 



2X27 
Corr.= - 



2292.43 

10X54 = _540 

1752.43 

3245 cub. yards = approx. vol. 



100X18 



12X6X27X94-67 
3245-91 = 3154 cub. yds. 



= — 91 cub. yds. 
== exact volume. 



94.67 



The above demonstration of the correction to be applied to 
the approximate vohime, found by averaging end areas, is intro- 
duced mainly to give an idea of the amount of that correction. 
Absolutely level sections are practically unknown, and the error 
involved in assuming any given sections as truly level will 
ordinarily be greater than the computed correction. If greater 
accuracy is required, more points should be obtained in the 
cross-sectioning, which will generally show that the sections 
are not truly level. 

77. Equivalent sections. When sections are very irregular 
the following method may be used, especially if great accuracy 
is not required. The sections are plotted to scale and then a 
uniform slope line is obtained by stretching a thread so that the 
undulations are averaged and an equivalent section is obtained. 
The center depth (d) and the slope angle (a) of this line can be 
obtained from the drawing, but it is more convenient to measure 
the distances (xi and Xr) from the center. The area may then 
be obtained independent of the center depth as follows: Let 

s=the slope ratio of the side slopes = cot /? = — . (See Fig. 50.) 



Then the 



Area 



^1 / Xl + Xr \ 

2\ s ') 



(Xl + Xr) 



Xr Xr 
~~S 2 " 



2a 

XI XI 

'~s2 



ah 



XlXr 
S 



oh_ 
' 2 ' 



(54) 



86 



KAILROAD CONSTRUCTION, 



§78. 



The true volume, according to the prismoidal formula, of a 
length of the road measured in this way will be 

iTxiW ah ( xi'-\-xrx/-\-Xr'[ 1 ab\ xi^'xr'' _ah'l. 
GLs 2"^V2 2 s 2 1'^ s 2J 

If computed by averaging end areas, the approximate volume 
will be 

iVxi'xr' ah xYW ahl 

2\_ s 2 '^ s 2 J 

Subtracting this result from the true volume, we obtain as the 
correction 

Correction = — (a;/'— a;/)(a;/— a:/'). . • (55) 



» 



This shows that if the side distances to either the right or 
left are equal at adjacent stations the correction is zero, and 
also that if the difference is small the correction is also small 
and very probably within the limit of accuracy obtainable by 
that method of cross-sectioning. In fact, as has already been 
jBhown in the latter part of § 75, it will usually be a useless 




Fig. 50. 

refinement to compute the prismoidal correction when the 
method of cross-sectioning is as rough and approximate as this 
method generally is. 

78. Equivalent level sections. These sloping " two-level " 
sections are sometimes transformed into " level sections of equal 



§ 78. EARTHWORK. 87 

area/' and the volume computed by the method of level sections 
(§ 75). But the true volume of a prismoid with sloping ends 
does not agree with that of a prismoid with equivalent bases and 
level ends except under special conditions, and when this method 
is used a correction must be applied if accuracy is desired, 
although, as intimated before, the assumption that the sections 
have uniform slopes will frequently introduce greater inaccuracies 
than that of this method of computation. The following dem- 
onstration is therefore given to show the scope and limitations 
of the errors involved in this much used method. 

In Fig. 50, let dj^ be the center height which gives an equiva- 
lent level section. The area will equal {a + d^ys — ^ , which 

must equal the area given m § 77, ~~^' ^^o~- 

.-. (a + d,ys='^\ 

or a + d,^^^^ (56) 

To obtain d, directly from notes, given in terms of d and a, 

we may substitute the values of xi and Xr given in § 74, which 

gives _ 

, , / , 7x tan /? a-\-d ,^^. 

a-\-di = (a+d) ^ ■= — . . (57) 

Vtan^ /?— tan^ a v 1 — s^ tan^ a 

The true volume of the equivalent section may be repre- 
sented by 

From this there should be subtracted the volume of the 
''grade prism" under the roadbed to obtain the volume of the 
cut that would be actually excavated, but in the following com- 
parison, as well as in other similar comparisons elsewhere made, 
the volume of the grade prism invariably cancels out, and so for 
the sake of simplicity it will be disregarded. This expression 
for volume may be transposed to 

h [ xiW {Vxi'x/ , \/xi"Xr"\ 2 , Xl'^Xr" "1 



88 RAILROAD CONSTRUCTION. § 79. 

The true volume of the prismoid with sloping ends is (see § 77) 

I Vxi'x/ , . / fxi' ^xr\ [x/-{-Xr'\ 1 \ , XlTx^'li 

The difference of the two volumes 

=-^{xi^Xr -^-Xj'x/ +Xl^Xr' ■\-Xi"Xr'—Xi'Xr' 

-2\/xVx/xt'W-xrx/') 
=^^(\/xi'xr''-\/xi''x/y ., (58) 



This shows that "equivalent level sections" do not in 
general give the true volume, there being an exception when 
xi'xr" ='Xi"xr , This condition is fulfilled when the slope is 
uniform, i.e., when a' = a'\ When this is nearly so the error 
is evidently not large. On the other hand, if the slopes are in- 
clined in opposite directions the error may be very considerable, 
particularly if the angles of slope are also large. 

79. Three-level sections. The next method of cross-section- 
ing in the order of complexity, and therefore in the order of 



Fig. 51. 

accuracy, is the method of three-level sections. The area o! 
the section is ^{a-\-d)(wr-Vwi) — ~j which may be written 



§ 79. EARTHWORK. 8& 

i(a + d)w — --', in which w=tVr+wi If the volume is com* 
puted by averaging end areas, it will equal 

j[(a+d')w'-ah + {^a+d'')w''-ahl . , , (59) 

If we divide by 27 to reduce to cubic yards, we have, when 
Z = 100, 

Vol f, )=ma + d')w'-i^ah + il(a + d")w'''-^ah 

For the next section 

Yolo,.. ^.j=||(a + d'Otf^"-|fa& + |K« + ^'"K"-tf^^ 
For a partial station length compute as usual and multiply 

result by — ^ — The prismoidal correction may be 

obtained by applying Eq 46 to each side in turn For the left 
side we have 

— -[(a + d^0-(« + O](^/'~^/), which equals 
For the right side we have, similarly, 

l^{d'-d"){Wr"-Wr'). 

The total correction therefore equals 

l^{d'-d"){{wi"+Wr") -{wi'-\-Wr')\ 
=-^{d'-d"){w"-w'). 

Reduced to cubic yards, and with Z = 100, 

Pris. Corr.=||(d'-d'0(^"-i^'). • . • (60) 

When this result is compared with that given in Eq. 55 there 
is an apparent inconsistency. If two-level ground is considered 
as but a special case of three -level ground, it would seem as if 



90 RAILROAD CONSTRUCTION. § 79. 

the same laws should apply. If, in Eq. 55, x/ =Xr\ and xi^' 
is different from Xi\ the equation i educes to zero; but in this 
case d' would also be different from c?''; and since xi' -\-Xr 
would =k;', and xi" ^x/^ =w'' in Eq. 60, w" —w' would not 
equal zero and the correction would be some finite quantity and 
not zero. The explanation lies in the difference in the form 
and volume of the prismoids, according to the method of the 
formation of the warped surfaces. If the surface is supposed to 
be generated by the locus of a line moving parallel to the ends 
as plane directors and along two straight lines lying in the side 
slopes, then xf^^^- will equal K^/ + ^/0? and a:r"^Jd. ^y]n equal 
^{xr +Xr'), but the profile of the center hne will not be straight 
and d^^^- will not equal J(^' + ^")- On the other hand, if the 
surfaces be generated by two lines moving parallel to the ends 
as plane directors and along a straight center line and straight 
side lines lying in the slopes, a warped surface will be generated 
each side of the center line, which will have uniform slopes on 
each side of the center at the two ends and nowhere else. This 
shows that when the upper surface of earthwork is warped (as 
it generally is), two-level ground should not be considered as a 
special case of three-level ground. This discussion, however, 
is only valuable to explain an apparent inconsistency and error. 
The method of two-level ground should only be used when 
such refinements as are here discussed are of no importance as 
affecting the accuracy. 

An example is given on the opposite page to illustrate the 
method of three-level sections. 

In the first column of yards 

210=tl(a + d)2^=f|X7.3X31.1; 
507, 734, etc., are found similarly; 
595=210-61+507-^61; 
448 =AW507 -61 + 734-61); 
602 = iVo(734-61 +392-61); 
449=392-61+179-61. 

For the prismoidal correction, 

-20=ff(rf'-d'OCi^''-^0=ff(2.6-8.1)(42.8-31.1) 
=ff(-5.5)( + 11.7). 

For the next line, -3 = xV^[|f(-2.6)( + 8.7)], and similarly 
for the rest. The "2^" in the columns of center heights, as welJ 



§79. 



EARTHAVORK. 



91 



> 


*^ 






rj< 


"^ 


iO 


CO 


II 




+ 


+ 


+ 


+ 






































<? 

^ 


CO 


r-l 


CO 


o 


Ol 


tH 


4- 


+ 


+ 


+ 


-r 














li^ 
















i> 


CD 


^ 


a 


Tt< 


'^ 


00 


CO 


t^ 


00 


H 


T-l 




<M 


tH 




m «-*' 




O 


CO 


tH 


CO 






(M 








u O 












PkU 




1 


1 


1 


1 


*C) 




t^ 


1> 


-^ 


^ 


1 




1-1 


00 


CO 


JO 














s 




+ 


+ 


1 


1 






lO 


o 


CO 


1> 


'ta 












1 




L'^ 


(N 


^ 


(M 






1 


i 


+ 


+ 


'^ 
















»o 


00 


(M 


a 






Oi 


rt< 


o 


t: 


13 




lO 


-^ 


CO 


rf^ 














U 












PH 


O 


1> 


TJ< 


<M 


Ci 


<M 


o 


C-0 


o 


1> 


»o 


T^ 


CO 


^ 


r^ 


00 


lO 


tH 


o 


^ 


M 


^ 


00 


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CO 


^ 


lO 


CO 


(N 


^ 


CO 


00 


■^^ 


y-i 


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+ 


1> 


(N 


lO 


^_^ 


00 


e 








tH 




i 


B;. 


oq 


&, - 


fe, oq 


fe. -^ 


fe. CO 


s 


00 

c 


00 


CO '"' 


00 ^ 








B^ 


35 


fen't^ 


^ ro 


^ 


o 


Bh 


t^ 




CC 




00( • 


(M • 


O 




or, 




^ 




fN 


•lo 


• t^ 




00 




lO 


l-^l 


C' 


CM 


lO:C0 


C CO 


rh 


(N 


lO 






'"' 


T-H 1 


CI 


^ 






fe, 


&, 


fe. 


fe. 


fe. 


g 


o 


'-^ 


I> 


-^ 


t- 


c^ 


00 


o 


CO 


CO 


o 















r^ 


00 


o 


a 


o 




1-H 


I-l 


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t-l . 






o 



t* 

^ 



Siw 






92 RAILROAD CONSTRUCTION. § 80. 

as in the columns of "right" and "left/' are inserted to indicate 

fill for all those points. Cut would be indicated by " C/' 

25 
8o. Computation of products. The quantities ^{a-\-d)w 

25 

and }z^dh represent in each case the product of two variable 

terms and a constant. These products are sometimes obtained 

from tables which are calculated for all ordinary ranges of the 

variable terms as arguments. A similar table computed for 

25 

'—~{d'—d"){w"—w') will assist similarly in computing the 

prismoidal correction. Prof. Charles L. Crandall, of Cornell 
University, is believed to be the first to prepare such a set of 
tables, which were first published in 1886 in "Tables for the 
Computation of Railway and Other Earthwork.'' Another 
easy method of obtaining these products is by the use of a slide- 
rule. A slide-rule has been designed by the author to accom- 
pany this volume.* It is designed particularly for this special 
work, although it may be utilized for many other purposes for 
which slide-rules are valuable. To illustrate its use, suppose 
{a-\-d) =28.2, and -u; = 62.4; then 

25 , ,, 28.2X62.4 
27^^ + ^)^^~T08^- 

Set 108 (which, being a constant of frequent use, is specially 
marked) on the sliding scale {B) opposite 282 on the other scale 
(A), and then opposite 624 on scale B will be found 1629 on 
scale A, the 162 being read directly and the 9 read by estima- 
tion.. Although strict rules may be followed for pointing off 
the final result, it only requires a very simple mental calculation 
to know that the result must be 1629 rather than 162.9 or 
16290. For products less than 1000 cubic 3^ards the result 
may be read directly from the scale; for products between 1000 
and 5000 the result may be read directly to the nearest 10 

* The first edition of this book was octavo, and a pasteboard sHde-rule, 
especially marked, accompanied each volume. Cutting down the size of 
^he pages to "pocket size " prevents the incorporation of the rule with the 
present edition. Any shde-rule with a logarithmic unit 22^ inches long will 
do equally well provided that the io8 mark is specially distinguished for 
ready use in computing the volume and that the 324 mark is similarly 
distinguished for use in computing the prismoidal correction. 



§81. EARTHWORK. 93 

yards, and the tenths of a division estimated. Between 5000 and 
10000 yards the result may be read directly to the nearest 20 
yards, and the fraction estimated; but prisms of such volume 
will never be found as simple triangular prisms — at least, an as- 
sumption that any mass of ground was as regular as this would 
probably involve more error than would occur from faulty esti- 
mation of fractional parts. Facilities for reading as high as 
10000 cubic yards would not have been put on the scale ex- 
cept for the necessity of finding such products as |f(9.1X9.5), 
for example. This product would be read off from the same 
part of the rule as f| (91X95). In the first case the product 
(80.0) could be read directly to the nearest .2 of a cubic yard, 
which is unnecessarily accurate. In the other case, the prod- 
uct (8004) could only be obtained by estimating /q of a division. 
The computation for the prismoidal correction may be made 
similarly except that the divisor is 3.24 instead of 1.08. For 

example, ^(5.5X11.7)= ^'^^^}'^ , Set the 324 on scale B 

(also specially marked like 108) opposite 55 on scale Aj and 
proceed as before. 

8i. Five-level sections. Sometimes the elevations over each 
edge of the roadbed are observed when cross-sectioning. These 
are distinctively termed ^'five-level sections." If the center, 
the slope-stakes, and one intermediate point on each side (not 
necessarily over the edge of the roadbed) are observed, it is 
termed an '^ irregular section." The field-work of cross-section- 
ing five-level sections is no less than for irregular sections T\nth 
one intermediate point; the computations, although capable of 
peculiar treatment on account of the location of the intermediate 
point, are no easier, and in some respects more laborious; the 
cross-sections obtained will not in general represent the actual 
cross-sections as truly as when there is perfect freedom in locat- 
ing the intermediate point ; as it is generally inadvisable or un- 
necessary to employ five-level sections throughout the length of 
a road, the change from one method to another adds a possible 
element of inaccuracy and loses the advantage of uniformity of 
method, particularly in the notes and form of computations. 
On these accounts the method will not be further developed, 
except to note that this case, as well as any other, may be 
solved by dividing the whole prismoid into triangular prismoids, 
computing the volume by averaging end areas, and computing 



94 



RAILROAD CONSTRUCTION. 



§82. 



the prismoidal correction by adding the computed corrections 
for each elementary triangular prismoid. 

82. Irregular sections. In cross-sectioning irregular sections, 
the distance from the center and the elevation above "grade" 
of every "break'' in the cross-section must be observed. The 
area of the irregular section may be obtained by computing the 
area of the trapezoids {fivey in Fig. 52 and subtracting the two 
external triangles. For Fig. 52 the area would be 



hl-\-h. . kl + d d^-jr ,ir + Jcr. . 



2 

kr-\-hr 



. hi/ h\ hr/ h\ 

^r-yr)-^{xi--j--i^Xr--^y 




Fig. 52. 



Expanding this and collecting terms, of which many will 
cancel, we obtain 



Area = — xiki-{-yi(d—hi) -{-Xrlcr-hyAJr—hr) 



L 



'\-Zr(d—kr)+ (hl+hr) 



]• 



(61) 



An examination of this formula will show a perfect regu- 
larity in its formation which will enable one to write out a 
similar formula for any section, no matter how irregular or how 



Ji 



§ 83. EARTHWORK. ^ 95 

many points there are, without any of the preliminary work. 
The formula may be expressed in words as follows : 

Akea e(^uals one-half the sum. of products obtained as follows : 

the distance to each slope-stake times the height above grade of 
the point next inside the slope-stake; 

the distance to each intermediate point in turn times the height of 
the point just inside minus the height of the point just outside; 

finally, one-half the ividth of the roadbed times the sum of the 
slope-stake heights. 

If one of the sides is perfectly regular from center to slope- 
stake, it is easy to show^ that the rule holds literally good. The 
'^ point next inside the slope- stake" in this case is the center; 
the intermediate terms for that side vanish. The last term 
must always be used. The rule holds good for three-level sec- 
tions, in w^hich case there are three terms, w^hich may be reduced 
to tw^o. Since these two terms arc both variable quantities for 
each cross-section, the special method, given in § 78, in which 

one term ( — j is a constant for all sections, is preferable. In 

the general method, each intermediate "break'' adds another 
term. 

83. Volume of an irregular prismoid. This is obtained by 
computing first the approximate volume by '^averaging end 
areas" or by multiplying the length by the half sum of the end 
areas, as computed from Eq. (61). In other words, the Approx. 

volume = 15;^ X 7^ (area' + area")- But since each area equals 

one-half the sum of products of width times height (see Eq. (61)) 
we may say that 

25 

Approx. volume = ^ (summation of width times height) . (62) 

the terms of width times height being like those found within 

the bracket of Eq. (61). 

As before, for partial station lengths, multiply the result by 

(length in feet -^ 100). There will be no constant subtractive 

25 
term, — ab, as in § 79. 

The correction to this approximate volume is found by 
considering that for the purpose of this correction ^nly the end 
sections are considered as 'Hhree-level" sections and the cor- 
rection is computed by applying Eq. (60). 



96 



RAILROAD CONSTRUCTION. 



§84. 



84. Numerical example; irregular sections; volume with 

approximate prismoidal correction. Assume the earthwork 

notes as given below, where the roadbed is 18 feet wide in cut 

and the slope is IJ to 1. Note that the stations read up the 

page and that when the surveyor is looking ahead along the 

line the several combinations of heights and distances out have 

approximately the same relative position on the note book as 

they have on the ground. For example, beginning at the 

8 9c 
bottom line (Sta. 16) the combination ^^' means that the 

extreme left-hand point of that section (the '^ slope stake '0 
is 22.4 feet horizontally from the center and that it is 8.9 feet 
above the required roadbed. The cut (c) would he 8.9 feet 
to reach the roadbed^ but of course the actual cutting is zero 
at the slope stake. The next point is 12.0 feet horizontally 
from the center and 7.6 feet above the road])ed. The cut at 
the center is 6.8 feet. The combinations of dimensions on the 
right-hand side are to be interpreted similarly. 



Sta. 



( cut 

Centers or 

(fill. 



Left. 



Right. 



19 
18 
17 
+ 42 
16 



0.6c 
2.3c 
7.6c 
10.2c 
6.8c 



3.6c 
14.4 

4.2c 



22.4 



6.8c 



3.2c 



15.3 


8.4 


5.2 


8.2c 


10.2c 


8.0c 


21.3 


17.4 


6.1 


12. 2c 




12 6c 


27.3 




8.2 


8.9c 




7.6c 



12.0 



O.lc 


0.4c 


4.2 


9.6 




1.2c 




10.8 




4.2c 




15.3 


6.2c 


8.4c 


7.5 


21.6 


3.2c 


2.6c 



4.1 



12.9 



The numerical computation is greatly facilitated by a sys- 
tematic form as given below. For Sta. 16, the first term is 
'Hhe distance to the left slope stake ^' (22.4) times ''the height 
above grade of the point next inside" (the height being 7.6), 
and we place this pair of figures in the columns of 'Svidth" 
and ''height." The "distance to the point next inside" is 
12.0 and the "height of the point just inside (6.8) minus the 



§85. 



EARTHWORK. 



97 



height of the point just outside" (8.9) equals ( — 2.1) and these 

25 
are the next pair of widths and heights. Taking — of the 

product of each pair of numbers we have the numbers in the 

first column of '^ yards.'' The sum of all these numbers in the 

42 
first and second groups multiplied by —-r (that section being 

only 42 feet long) equals 378 cubic yards, the volume by averag- 
ing end areas. The determination of center heights and total 
widths and the application of Eq. (60), to obtain the approxi- 
mate prismoidal correction, is self-evident. 





VOLUME OF IRREGULAR PRISMOID, WITH APPROXIMATE 








PRISMOIDAL CORRECTION. 


Sta. 


W'th 


H'ght 


Yards. 


Cen. 
Height. 


Total 
width 

35.3 


d'-d" 


w'^-w' 


Approx. 
pris.corr. 




22.4 


7.6 


158 




+ 6.8 










12.0 


-2.1 


-23 














16 


12.9 
4.1 
9.0 


3.2 

4.2 
11.5 


40 
16 
96 
















27.3 


12.6 


319 




+ 10.2 


48.9 


-3.4 


+ 13.6 


-14 




8.2 


-2.0 


-15 














+ 42 


21.6 
7.5 


6.2 

1.8 


124 
13 
















9.0 


20.6 


172 


378 


„ 








(-6) 




21.3 


10.2 


201 




+ 7.6 


36.6 


+ 2.6 


-12.3 


-10 




17.4 


-0.2 


- 3 














17 


6.1 
15.3 


-2.6 
7.6 


-14 

107 
















9.0 


12.4 


103 


584 










(-6) 




15.3 


6.8 


95 




+ 2.3 


26.1 


+ 5.3 


-10.5 


-17 




8.4 


-1.0 


- 7 














18 


5.2 
10.8 


-4.5 
2.3 


-22 
23 
















9.0 


5.4 


45 


528 










(-17) 




14.4 


0.6 


8 




+ 0.6 


24.0 


+ 1.7 


-2.1 


-1 


19 


9.6 


0.1 


1 














4.2 


0.2 


1 
















9 


4.0 


33 


177 










(-1) 



-30 



Approx. volume =1667 
Approx. pris. corr. = — 30 

Corrected volume = 1637 cubic yards 

85. Magnitude of the probable error of this method. In 
previous editions of this work, methods were given for com- 
puting the mathematically exact volume of a prismoid whose 
ends coincide with the ^'irregular sections" as measured, and 



98 RAILROAD CONSTRUCTION. § 85. 

whose upper surfaces are assumed to coincide with the actual 
surface of the ground. As in the previous methods, the ^'ap- 
proximate volume" is computed by averaging end areas and 
then a correction is appUed. If the end sections have the same 
number of intermediate points on each side, and if it can be as- 
sumed that the corresponding Hues in each section are connected 
by plana or warped surfaces, which coincide with the surface of 
the ground, then the mathematically exact or *Hrue'' correc- 
tion may be obtained by dividing the volume into elementary 
triangular prismoids, finding the correction for each and adding 
the results. Although such a method appears very complicated, 
it is readily possible to develop a law by means of which the 
true prismoidal correction may be written out (similarly to 
writing out the formula for the area, Eq. (61)) without any 
preliminary calculation. Such a law has a mathematical 
fascination, but it should be remembered that when the ground 
surface is so broken up that the cross-sections are '' irregular" 
it is in general correspondingly rough and irregular between 
the cross-sections, especially when those sections are 100 feet 
apart. It is also true that the cross-sections do not usually 
have the same number of intermediate points on corresponding 
sides of the center. In such a case, unless the actual form of 
the ground between the cross-sections is observed and measured, 
the exact method cannot be used. An extra point in one cross- 
section implies an extra ridge (or hollow) which ''runs out" 
or disappears by the time the adjoining section is reached. 
Theoretically a cross-section should be taken at the point where 
such a ridge or hollow runs out. In general this point will not 
be at an even 100-foot station. The attempt to compute the 
exact prismoidal correction usually gives merely a false appear- 
ance of extreme accuracy to the work which is not justified 
by the results. It should not be forgotten that it is readily 
possible to spend an amount of time on the surveying and 
computing which is worth more than the few cubic yards of 
earth which represents the additional accuracy of the more 
precise method. The accuracy of the office computation should 
be kept proportionate to the accuracy of the cross-sectioning 
in the field. The discussion of the magnitude of the prismoidal 
correction in §§ 72-79 shows that it is small except when the 
two ends of the prismoid are very dissimilar. The dissimiliarity 
between the two ends of a prismoid would be substantially the 



83. 



EARTHWORK. 



99 



same whether the ends were actually ''irregular'' or had "three- 
lever' sections, which for each end had the same slope stakes 
and center heights as the irregular sections. Experience proves 
that the approximate prismoidal correction, computed by 
considering the ground as three-level, is so nearly equal to the 
true prismoidal correction that the difference is perhaps no 
greater than the probable difference between the true volume 
of earth and the volume of the geometrical prismoid which is 
assumed to represent that volume. The experienced surveyor 
will take his cross-sections at such places and so close together 
that the warped surfaces joining the sections will lie very nearly 
in the surface or at least will so average the errors that they 
will substantially neutralize each other. 

86. Numerical illustration of the accuracy of the approxi- 
mate rule. The ''true'' prismoidal correction for the numerical 
case given in § 84 was computed by the method outlined above, 
and on the basis of certain figiu-es as to the vanishing of the 
ridges and valleys found in one section and not found in the 
adjacent sections. The various quantities for the volumes 
between the cross-sections have been tabulated as shown. 





1 


2 


3 


4 


5 


6 


7 


Sections. 


p CU C3 


ill 

m 


'o 

i 


Approx. pris. 

COIT. on basis 

of three-level 

ground. 




Approx. vol. 

computed 

from center 

and side 
heights only. 


CO 

gr 

Hco 


16 16 + 42 

16+42.. 17 

17 18 

IS 19 


378 
584 
528 
177 


- 5 

- 3 

-16 

- 3 


373 

581 
512 
174 


- 6 

- 6 
-17 

- 1 


-1 

-3 
-1 

+ 2 


396 
577 
463 
147 


-23 

+ 4 
+ 49 

+ 27 




1667 


-27 


1640 


-30 


-3 


1583 


+ 57 



There has also been shown in the last two columns the error 
involved if the "intermediate points" had been ignored in 
the cross-sectioning. From the tabular form we may learn that 

1. The differences between the "true" and approximate 
corrections is so small that it is probably swallowed up by errors 
resulting from inaccurate cross-sectioning. 

2. The error which would have been involved in ignoring 
the intermediate points is so very large in comparison with 



100 RAILROAD CONSTRUCTION. § 87. 

the other corresponding errors that (although it proves nothing 
absolutely definite, being an individual case) the probabilities 
of the relative error from these sources are clearly indicated. 

87. Cross-sectioning irregular sections. The slope stake 
should preferably be determined first, and then the ^^ breaks" 
between the slope stake and the center. When, as is usual, 
the ground is not even between the cross-section just taken 
and the section at the next 100-foot station, a point should be 
selected for a cross-section such that the lines to the previous 
section should coincide with the actual surface of the ground as 
closely as the accuracy of the work demands. § 94 gives 
a numerical illustration of the magnitude of some of these 
errors. Although it is possible for a skillful surveyor to so 
choose his cross- sections in rough and irregular ground that 
the positive and negative errors will nearly balance, it requires 
exceptional skill. Frequently the work may be simplified by 
computing separately the volume of a mound or pit, the 
existence of which has been ignored in the regular cross- 
sectioning. 

88. Side-hill work. When the natural slope cuts the roadbed 
there is a necessitv for both cut and fill at the same cross-section. 



Fig. 53. 

When this occurs the cross-sections of both cut and fill are often 
so nearly triangular that they may be considered as such without 
great error, and the volumes may be computed separately as 
triangular prismoids without adopting the more elaborate form 
of computation so necessary for complicated irregular sections. 
When the ground is too irregular for this the best plan is to 



§88. 



EARTHWORK. 



101 



follow the uniform system. In computing the cut, as in Fig. 53. 
the left side would be as usual > there would be a small center 
cut and an ordinate of zero at a short distance to the right of the 
center. Then, ignoring the fill, and applying Eq. 61 strictly, 
we have two terms for the left side, one for the right, and the 
term involving J6, which will be ^hhi in this case, since hr=0, 
and the equation becomes 

Aiesi = i[xiki-hyi(d—hi) -{-Xrd + ^hhi]. 

The area for fill may also be computed by a strict application 
of Eq. 61, but for Fig. 54 all distances for the left side are zero 
and the elevation for the first point out is zero, d also must be 




considered as zero. Following the rule, § 81, literall}^, the equa- 
tion becomes 

ATe2i(Fm) =i[Xrkr + yr(0—hr) -]-Zr(0—kr) -\-ih(0 + hr)l 



which reduces to 



ilXrhr—yrhr — Zrkr + ihhr]. 



(Note that Xr, hr, etc., have different significations and values 
in this and in the preceding paragraphs.) The '' terminal 
pyramids" illustrated in Fig. 40 are instances of side-hill work 
for very short distances. Since side-hill work always implies 
both cut and fill at the same cross-section, whenever either the 
cut or fill disappears and the earthwork becomes wholly cut or 
wholly fill, that point marks the end of the "side-hill work,'' 
and a cross-section should be taken at this point. 



102 RAILROAD CONSTRUCTION. § 89. 

89. Borrow-pits. The cross-sections of borrow-pits will vary 
not only on account of the undulations of the surface of the 




ground, but also on the sides, according to whether they are 
made by widening a convenient cut (as illustrated in Fig. 55) 
or simply by digging a pit. The sides should always be prop- 
Trly sloped and the cutting made cleanly, so as to avoid un- 
sightly roughness. If the slope ratio on the right-hand side 
(Fig. 55) is s, the area of the triangle is Jsm^ The area of the 
section is i[ug-\-(g + h)v + (h+j)x + (j + k)y + (k + m)z-sm^l If 
all the horizontal measurements were referred to one side as 
an origin, a formula similar to Eq. 61 could readily be devel- 
oped, but little or no advantage would be gained on accoimt of 
any simplicity of computation. Since the exact volume of the 
earth borrowed is frequently necessary, the prismoidal correc- 
tion should be computed; and since such a section as Fig. 55 
does not even approximate to a three-level section, the method 
suggested in § 83 cannot be employed. It will then be neces- 
sary to employ the more exact method of dividing the volume 
into triangular prismoids and taking the summation of their 
corrections, found according to the general method of § 72. 

90. Correction for curvature. The volume of a solid, gen- 
erated by revolving a plane area about an axis lying in the 
plane but outside of the area, equals the product of the given 
area times the length of the path of the center of gravity of the 
area. If the centers of gravity of all cross-sections lie in the 
center of the road, where the length of the road is measured, 
there is absolutely no necessary correction for curvature. If all 
the cross-sections in any given length were exactly the same and 
therefore had the same eccentricity, the correction for curvature 
would be very readily computed according to the above prin- 
ciple. But when both the areas and the eccentricities vary 
from point to point, as is generally the case, a theoretically exact 



§ 90. EARTHWORK. 103 

solution is quite complex, both in its derivation and application. 
Suppose, for simplicity, a curved section of the road, of uniform 
cross-sections and with the center of gravity of every cross- 
section at the same distance e from the center line of the road 
The length of the path of the center of gravity will be to the 
length of the center line as R±e : R. Therefore we have 

R:iie 

True vol.: nominal vol. :: Rdie '. R. /. True vol.=lA for 

K 

a volume of uniform area and eccentricity. For any other area 

R±e^ 
and eccentricity we have, similarly. True vol.' = IA' — =5 — . This 

shows that the effect of curvature is the same as increasing (or 
diminishing) the area by a quantity depending on the area and 
eccentricity, the increased (or diminished) area being found by 

multiplying the actual area by the ratio — ^— . This being 

K 

independent of the value of Z, it is true for infinitesimal lengths. 

If the eccentricity is assumed to vary uniformly between two 

sections, the equivalent area of a cross-section located midway 

('^±^') 

between the two end cross-sections would be Am— ^ » 

Therefore the volume of a solid w^hich, when straight, would be 
— (A'-l-4Aw4-A'0, w^ould then become 

D 

rwroL=^rA'(E±60+4A^(i^±5^)+A''(E±oT 

Subtracting the nominal volume (the true volume when the 
prismoid is straight), the 

Correction =±~^ (A' + 2A„,)e' + (2A^+A'0e'' L . (63) 

Another demonstration of the same result is given by Prof. 
C. L. Crandall in his ^'Tables for the Computation of Railw^ay 
and other Earthwork," in which is obtained by calculus methods 
the summation of elementary volumes having variable areas 
with variable eccentricities. The exact application of Eq. 63 
requires that A^ be known, which requires laborious computa- 



104 RAILROAD CONSTRUCTION. § 91. 

tions, but no error worth considering is involved if the equation 
is written approximately 

C2^r2;.corr.=2p(AV + A'V'), .... (64) 

which is the equation generally used. The approximation con- 
sists in assuming that the difference between A' and A^ equals 
the difference between Am and A'' but with opposite sign. The - 
error due to the approximation is always utterly insignificant. I 

91. Eccentricity of the center of gravity. The determination 
of the true positions of the centers of gravity of a long series of 
irregular cross-sections would be a very laborious operation, 
but fortunately it is generally sufficiently accurate to consider 
the cross-sections as three-level ground, or, for side-hill work, to 




Fig. 56. 



be triangular, for the purpose of this correction.. The eccentricity 
of the cross-section of Fig. 56 (including the grade triangle) may 
be written 



(a-\-d)xiXi (a-^d)XrXr 



O 1 Xi — Xr 1 / >, /r»r\ 

= -(Xi-Xr). . (65) 



(a-^-d^xi (a-^d)Xr 3 Xi-}-Xr 3 



The side toward xi being considered positive in the above 
demonstration, if Xr>xi, e would be negative, i.e., the center 
of gravity would be on the right side. Therefore, for three-level 



§ 91. EARTHWORK. 105 

ground, the correction for curvature (see Eq. 64) may be written 
Correction = ^[A\xf-x/) + A''(x/' -x/')]. 
Since the approximate volume of the prismoid is 

in which F' and V'^ represent the number of cubic yards corre- 
sponding to the area at each station, we may write 

Corr, in cub. yds. = ^IV'W-^r) + V'^x/' -a:/')]- • (66) 

It should be noted that the value of e, derived in Eq. 65, is 
the eccentricity of the whole area including the triangle under 
the roadbed. The eccentricity of the true area is greater than 
this and equals 

true area + hah 

eX — 7 ^— =^1. 

true area 

The required quantity (AV of Eq. 64) equals true areaXe^ 
which equals (true area + iah) X e. Since the value of e is very 
simple, while the value of e^ would, in general, be a complex 
quantity, it is easier to use the simple value of Eq. 65 and add 
Ja6 to the area. Therefore, in the case of three-level ground 
the subtractive term ff a?? (§ 79) should not be subtracted in 
computing this correction. For irregular ground, when com- 
puted by the method given in §§82 and 83, which does not 
involve the grade triangle, a term ^^ah m.ust be added at every 
station when computing the quantities F' and V'^ for Eq. 66. 

It should be noted that the factor 1-~SR, which is constant 
for the length of the curve, may be computed with all necessary 
accuracy and without resorting to tables by remembering that 

5730 



degree of curve' 



Since it is useless to attempt the computation of railroad 
earthwork closer than the nearest cubic yard, it will frequently 



106 



BAILROAD CONSTEUCTION. 



§91. 



be possible to write out all curvature corrections by a simple 
mental process upon a mere inspection of the computation sheet. 
Eq. 66 shows that the correction for each station is of the form 

V(Xi-Xr) 



SR 



SR is generally a large quantity — for a 6° curve 



it is 2865. (xi—Xr) is generally small. It may frequently be 
seen by inspection that the product V(xi—Xr) is roughly twice 
or three times SR, or perhaps less than half of 3R, so that the 
corrective term for that station may be written 2, 3, or cubic 
yards, the fraction being disregarded. For much larger absolute 
amounts the correction must be computed with a correspondingly 
closer percentage of accuracy. 

The algebraic sign of the curvature correction is best deter- 
mined by noting that the center of gravity of the cross-section is 
on the right or left side of the center according as a: r is greater 
or less than xi, and that the correction is positive if the center of 
gravity is on the outside of the curve, and negative if on the 
inside. 

It is frequently found that xi is uniformly greater (or uni- 
formly less) than Xr throughout the length of the curve. Then 
the curvature correction for each station is uniformly positive or 
negative. But in irregular ground the center of gravity is apt 




Fia. 57. 



to be irregularly on the outside or on the inside of the curve, 
and the curvature correction will be correspondingly positive or 
negative. If the curve is to the right, the correction will be 
positive or negative according as (xi—Xr) is positive or negative; 
if the curve is to the left, the correction will be positive or nega- 



§ 92. EARTHWORK. 107 

tive according as {xr—xj) is positive or negative. Therefore 
when computing curves to the right use the form {xi—Xr) in 
Eqs. 66 and 68; when computing curves to the left use the form 
{xr—xi) in these equations; the algebraic sign of the correction 
will then be strictl}^ in accordance wdth the results thus obtained. 
92. Center of gravity of side-hill sections. In computing the 
correction for side-hill work the cross-section would be treated 
as triangular unless the error involved would evidently be too^ 
great to be disregarded. The center of gravity of the triangle^ 
lies on the line joining the vertex with the middle of the base 
and at J of the length of this line from the base. It is therefore 
equal to the distance from the center to the foot of this line plus 
J of its horizontal projection. Therefore 

__h Xr XI h Xr 

'"4~"2"^I~T2 6 

h XI Xr 

'"6"^3~3 

=i[|+(..-..)]..,. . . (67) 

By the same process as that used in § 91 the correction equation 
may be written 

Corr.incub.yds. = 3^[7^(|-4CV-^r'>) + F^'(| + (rrz''-:r/0)]. (68) 

It should be noted that since the grade triangle is not used in 
this computation the volume of the grade prism is not involved 
in computing the quantities V and V", 

The eccentricities of cross-sections in side-hill work are never 
zero, and are frequently quite large. The total volume is gen- 
erally quite small. It follows that the correction for curvature 
is generally a vastly larger proportion of the total volume than 
in ordinary three-level or irregular sections. 

If the triangle is wholly to one side of the center, Eq. 67 can 
still be used. For example, to compute the eccentricity of the 
triangle of fill, Fig. 57, denote the two distances to the slope- 



108 RAILROAD CONSTRUCTION. § 93. 

stakes by yr and —yi (note the minus sign). Applying Eq. 67 
literally (noting that — must here be considered as negative in 
order to make the notation consistent) we obtain 



1 






-yr) , 



which reduces to 



^=-|-! |- + ^z + 2/r. (69) 



As the algebraic signs tend to create confusion in these 
formulae, it is more simple to remember that for a triangle 
lying on both sides of the center e is always numerically equal 

to~-i —-\-{xi'^Xt) L and for a triangle entirely on one side, e is 



'3-L2 



numerically equal to— ^ + the numerical sumoi the two dis- 
tances out]. The algebraic sign of e is readily determinable as 
in § 91. 

93. Example of curvature correction. Assume that the fill in 

§ 79 occurred on a 6° curve to the rtght, -— = . The 

oK 28d5 

quantities 210, 507, etc., represent the quantities V% F", etc.^ 

since they include in each case the 61 cubic yards due to the 

grade prism. Then 



7(a;;^:rr) ^ 210(22.9-8.2) ^ 3101.7 

SR ~ 2865 ~ 2865 ~ "^ * 



The sign is plus, since the center of gravity of the cross-sec- 
tion is on the left side of the center and the road curves to the 
right, thus making the true volume larger. For Sta. 18 the 
correction, computed similarly, is +3, and the correction for 
the whole section is 1+3=4. For Sta. 18 + 40 the correction 
is computed as 6 yards. Therefore, for the 40 feet, the correc- 
tion is tV(j(3 + 6) =3.6, which is called 4. Computing the others 
similarly we obtain a total correction of + 16 cubic yards. 



§ 94. EARTHWORK. 109 

94. Accuracy of earthwork computations. The precedmg 
methods give the precise volume (except where approximations 
are distinctly admitted) of the prismoids which are supposed to 
represent the volume of the earthwork. To appreciate the 
accuracy necessary in cross-sectioning to obtain a given accuracy 
in volume, consider that a fifteen-foot length of the cross-section, 
which is assumed to be straight, really sags 0.1 foot, so that the 
cross-section is in error by a triangle 15 feet wide and 0.1 foot 
high. This sag 0.1 foot high would hardly be detected by the 
eye, but in a length of 100 feet in each direction it would make 
an error of volume of 1.4 cubic yards in each of the two pris- 
moids, assuming that the sections at the other ends were perfect. 
If the cross-sections at both ends of a prismoid were in error by 
this same amount, the volume of that prismoid would be in error 
by 2.8 cubic yards if the errors of area were both plus or both 
minus. If one were plus and one minus, the errors would 
neutralize each other, and it is the compensating character of 
these errors which permits any confidence in the results as 
obtained by the usual methods of cross-sectioning. It demon- 
strates the utter futility of attempting any closer accuracy than 
the nearest cubic yard. It will thus be seen that if an error 
really exists at any cross-section it involves the prismoids on 
both sides of the section, evenTthough all the other cross-sections 
are perfect. As a further illustration, suppose that cross-sec- 
tions were taken by the method of slope angle and center depth 
(§ 74), and that a cross-section, assumed as uniform, sags 0.4 
foot in a width of 20 feet. Assume an equal error (of same 
sign) at the other end of a 100-foot section. The error of 
volume for that one prismoid is 38 cubic yards. 

The computations further assume that the w^arped surface, 
passing through the end sections, coincides with the surface of 
the ground. Suppose that the cross-sectioning had been done 
with mathematical perfection; and, to assume a simple case, 
suppose a sag of 0.5 foot between the sections, which causes an 
error equal to the volume of a pyramid having a base of 20 feet 
(in each cross-section) times 100 feet (between the cross-sec- 
tions) and a height of 0.5 foot. The volume of this pyramid is 
J(20X100)X0.5 = 333 cub. ft. = 12 cub. yds. And yet this sag 
or hump of 6 inches would generally be utterly unnoticed, or 
at least disregarded. 

When the ground is very rough and broken it is sometimes 



110 



RAILROAD CONSTRUCTION. 



§95. 



practically impossible, even with frequent cross-sections, to 
locate warped surfaces which will closely coincide with all the 
sudden irregularities of the ground. In such cases the compu- 
tations are necessarily more or less approximate and dependence 
must be placed on the compensating character of the errors. 

95. Approximate computations from profiles. When a 
''paper location'' has been laid out on a topographical map 
having contours, it is possible to compute approximately the 
amount of earthwork required by some very simple and rapid 
calculations. A profile may be readily drawn by noting the 
intersections of the proposed center line with the various con- 
tours and plotting the surface line on profile paper. Drawing 
the grade-line on the profile, the depth of cut or fill may be 
Bcaled off at any point. When it is only desired to obtain 




I 



Fia. 58. 

very quickly an approximate estimate of the amount of earth- 
work required on a suggested line, it may be done by the method 
described in § 75, or by the use of Table XXXIII. But the 
assumption that the surface of the ground at each cross-section K 
is level invariably has the effect that the estimated volumes w 
are not as large as those actually required. The difference 
between the '-^evel section" hkms and the actual slope section 
hknq equals the difference between the triangles mon and oqs, 
and this difference equals the shaded area mpn. The excess 
volume is proportional to the area of the triangle mpn. This 
area may be expressed by the formula, 



Area mpn ==2(yo + d cot /?) 



jsin^o: sin/? cos /9 
cos 2a: — cos 2^* 



I 



§ 95. EARTHWOEK. Ill 

The percentage of this excess area to the nominal area hJcma 
therefore depends on the dimensions h and d and the angles a 
and p. A solution of this equation for ninety different com- 
binations of various numerical values for these four variables 
is included in Table XXXIII for the purpose of making cor- 
rections. A study of this correction table points conclusively 
to the following laws, a thorough understanding of which will 
enable an engineer to appreciate the degree of accuracy which 
is attainable by this approximate method: 

(a) Increasing the ividth of the roadbed (5), the other three 
factors remaining constant, increases the percentage of error, 
but the increase is comparatively small. 

(b) Increasing the depth of cut or fill (c?), decreases the per- 
centage of error, but the decrease is almost insignificant. 

(c) Increasing the angle of the side slopes (,5) decreases the 
percentage of error, the decrease being very considerable. 

(d) Increasing the angle of the slope of the ground (a), 
increases the percentage of error, the percentage rapidly in- 
creasing to infinity as the value of ^ a approaches that of p. 
This is another method of stating the fact that a must always 
be less than /? and, practically, must be considerably less, so 
that the slope stake shall be within a reasonable distance from 
the center. - 

Since the above value for the corrective area is a function of 
the angle a, which is usually variable and whose value is fre- 
quently know^n only approximately, it is useless to attempt 
to apply the correction wdth great precision, and the following 
rules will usually be found amply accurate, considering the 
probable lack of precision in the data used. 

1. For embankments or cuts, having a slope of 1.5:1, and 
with a surface slope of 5° (nearly 9%) the excess of true area 
over nominal area is about 2%. There is only a slight varia- 
tion from this value for all ordinary depths (d) and widths (b) 
of roadbed. Therefore the nominal volume w^ould be about 2% 
too small. On the other hand, the effect of the prismoidal 
correction is such that, even with truly level sections, the 
nominal volume is too large. See §§75 and 76. The amount 
of the prismoidal correction depends on the differences between 
successive center depths. In the very ordinary numerical 
case given in § 76, the correction was nearly 3%, which more 
than neutralizes the error due to surface slope. Therefore in 



112 RAILROAD CONSTRUCTION. §95. 

many cases on slightly sloping ground the error due to the 
surface slope will so nearly neutralize the prismoidal correc- 
tion that the quantities taken directly from the tables (without 
correction for either cause) will equal the true volume with as 
close an approach to accuracy as the precision of the surveying 
will permit. 

2. For a cut with a slope of 1:1, and with a surface slope of 
5° the error is about 1%. This will be neutralized by still 
smaller prismoidal corrections. Therefore, for surface slopes 
of 5° or less, no allowance should be made for this error unless 
the prismoidal correction is also considered. 

3. When the surface slope is 10° (nearly 18%) the error for 
a 1.5:1 slope is from 7% to 10% and for a 1:1 slope from 3% 
to 5%. 

4. For a 30° surface slope and 1.5:1 side slopes the excess 
volume is three or four times the nominal volume. Such a 
steep surface slope implies the probability of ''side-hill work" 
to which the above corrective rules are not applicable. When 
the surface slopes are very steep careful work must be [done 
to avoid excessive error. For a 1 : 1 side slope, the errors are 
from 50%) to 80%. 

A still closer approximation, especially for the steeper surface 
slopes, may be obtained by using, directly or by interpolation, 
figures from the corrective tabular form which forms part of 
Table XXXIII. Unless the surface slope angle is known 
accurately (especially when large) no great accuracy in the 
final result is possible. Close accuracy would also require the 
determination of the prismoidal correction. But if such close 
accuracy is deemed essential, it can be most easily obtained 
by accurate cross-sectioning at each station and the adoption 
of other methods of computation — such as are given in §§83 
and 84. 

When the contours have been drawn in for a sufficient 
distance on either side to include the position of both slope 
stakes at every station, as will usually be the case, cross-sections 
may be obtained by drawing lines on the map at each station 
perpendicular to the center line — see Fig. 4. The intersection 
of these lines with the contours will furnish the distances for 
drawing on cross-section paper the transverse profile at each 
station. Drawing on the same cross-section the lines repre- 
senting the roadbed and the side slopes, the cross-section of 



§ 95. EARTHWORK. 113 

cut (or fill) is complete and its area may be obtained by scaling 
from the cross-section paper. If the contours have been 
located on the map with sufficient accuracy, such a method 
will determine the cross-sectional area very closely. When 
cross-sections have been taken with a wye- or hand-level, as 
described in § 12, the cross-sections as plotted will probably 
be more accurate than when the contours are run in from 
points determined by the stadia method. In fact this semi- 
graphical method is frequently used, in place of the purely 
numerical methods described in previous sections, to make 
final estimates of the volume of earthwork. 

As a numerical example, an assumed location line was laid 
out on the contours given in Fig. 4. The volume of cut, as 
determined by Table XXXIII for a roadbed 20 feet wide, with 
side slopes of 1:1, was 5746 cubic yards. The surface slope 
varied from 3° to 11°. Computing the corrections by a careful 
interpolation from the corrective table, the total correction was 
found to be 128 cubic yards, or an average of a little over 2%. 
On the other hand the negative prismoidal correction amounts 
to 72 cubic yards, which leaves a net correction of 56 cubic 
yards — about 1%. It so happens that in this case a correction 
for curvature would tend further to wipe out this correction. 
These figures merely verify numerically the general conclusions 
stated above, although it should not be forgotten that in indi- 
vidual cases the figures taken from Table XXXIII require 
ample correction. 

The following approximate rule, for which the author is 
indebted to Mr. W. H. Edinger, is exceedingly useful when it 
is desired to rapidly determine the approximate volume of 
earthwork between two points along the road. Its great merit 
lies in the fact that it only means the memorizing of a com- 
paratively simple rule which will make it possible to make 
such computations in the field, without the use of tables. The 
rule is based on the fact that the area of any level section equals 
bd + sd^ ; and therefore, 

S(vol.) = (6S(^+sSd2)^, 

in which L is usually 100 feet. For strict accuracy this would 
only be the volume provided the total length was an even num- 
ber of hundred feet, and the various values of d represented 



114 RAILBOAD CONSTRUCTION. § 96 

the depths which were uniform for hundred foot sections. It 
makes no allowance for the comparatively large prismoidal 
error of the pyramidal and wedge-shaped sections usually found 
at each end of a cut or fill, but where an approximate estimate 
is desired, in which this inaccuracy may be neglected, the 
method is very useful. The method of applying this rule with- 
out tables may best be illustrated by a simple numerical ex- 
ample. Assume that the levels on a stretch of fairly level 
ground, which is about 500 feet long, have been taken, the depths 
being taken at points 100 feet apart, the first and last points 
being about 40 or 50 feet from the ends of the cut, or fill. The 
depths are as given in the first column in the tabular form 
below; the slope is 1.5:1, and the breadth (6) is 14 feet. 



d 


d' 


1.6 


2.56 


2.8 


7.84 


4.5 


20.25 


3.1 


9.61 


0.9 


.81 


ScZ- 12.9 


2^2=41.07 


14 


20.53 


6ScZ = 180.6 


SSd2_ei gQ 


61.60 




242.2 




24220 -f 


-27 = 897 cubic yards. 



The 180.6 is the hXd and the 61.6 is sSc?^; adding these and 
moving the decimal point two places to multiply by 100, we 
only have to divide by 27 to obtain the value in cubic yards. 
Although the above rule requires more work than the employ- 
ment of earthwork tables, yet it is a very convenient method 
of estimating the approximate volume of a short section of 
earthwork when no tables are at hand. 

96. Shrinkage of earthwork. The statistical data indicating 
the amount of shrinkage is very conflicting, a fact which is 
probably due to the following causes: 

1. The various kinds of earthy material act very differently 
as respects shrinkage. There is a great lack of uniformity in 



§ 96. EARTHWORK. 115 

the classification of earths in the tests and experiments which 
have been made. 

2. Very much depends on the method of forming an embank- 
ment (as will be shown later). Different reports have been 
based on different methods — often without mention of the 
method. 

3. An embankment requires considerable time to shrink to 
its final volume, and therefore much depends on the time 
elapsed between construction and the measurement of what is 
supposed to be the settled volume. 

4. A soft subsoil will frequently settle under the weight of a 
high embankment and apparently indicate a far greater shrink- 
age than the actual reduction in volume. 

5. An embankment of very soft material will sometimes 
"mush^* or widen at the sides, with a consequent settling of 
the top, due to this cause alone. 

This subject has called forth much discussion in the technical 
press and literature. Quotations can be made of figures cover- 
ing a large range of values, but space will only permit the 
statement of the conclusions which may be drawn from the 
large mass of testimony which has been presented. 

1. Volume of loose material. "When material of any character 
is excavated and deposited loosely in a pile, its volume is 
always largely in excess of the volume of the excavation. 
Solid rock will occupy from 60% to 80% more space when 
broken up than when solid. A soft earth will have an excess 
volume of about 20% to 25%. 

2. Effect of method of depositing. "When material is de- 
posited loosely, as from a trestle, the excess of volume when 
the embankment is just completed is very large. The time 
required for final settlement is also very great. When an 
embankment is formed by the wheelbarrow method, the initial 
expansion is about as great as when the material is merely 
dumped from cars. When the material is deposited in small 
increments from wagons and each layer is subjected to com- 
pression from horses' hoofs and from wheels, the contraction 
during construction is far greater and the additional shrinkage 
is comparatively small. Wheeled scrapers and drag scrapers 
will produce even more initial compression. 

3. Time required for final settlement. This depends partly 
on the method of formation and also on the character of the 



116 RAILROAD CONSTRUCTION. §97. 

material. When a soft loamy soil is deposited loosely, the dry- 
ing out of the soil during the first long dry season will develop 
large cracks. Subsequent rains will close these cracks by a 
general contraction of the whole mass. When the embank- 
ment is loosely formed it may take two years before additional 
settlement becomes inappreciable, but when the method of 
deposition ensures compression during construction the subse- 
quent shrinkage is less in time as well as amount. 

4. Classification of soils with respect to shrinkage. Loose 
vegetable surface soil will expand very greatly when excavated 
and first deposited, but will subsequently shrink to considerably 
less than its original volume. Clay soils are next in order 
and the sandy and gravelly soils come at the other end of 
the list of earthy materials. Rock expands very greatly when 
first broken up and deposited and there is no appreciable sub- 
sequent shrinkage. 

97. Proper allowance for shrinkage. Specifications for the 
Mississippi River levees require that there shall be a 10% 
shrinkage allowance for embankments formed by team work 
and 25% allowance for wheelbarrow work. It is contended 



Fia. 59. 

that such figures are only justified because the subsoil settles 
or because the embankments mush out at the sides, and that 
if these effects do not occur the levees are permanently higher 
than designed. 

It is usual to require that embankments shall be constructed 
higher than their desired ultimate, as shown in Fig. 59. Since 
the base does not contract, the contraction may be said to 
be all vertical. Since a high embankment will unquestionably 
shrink a greater total amount than a low embankment (what- 
ever the percentage), it follows that an embankment having 



§ 97, EARTHWORK. 117 

variable heights (as usual) should have an initial grade-line 
somewhat like the dotted line adc in Fig. 60. Although some 
such method is essential if there is to be no ultimate sag below 
the desired grade-line, the policy is sharply criticized. The 
grade ad, even though temporary, may prove objectionable 
from an operating standpoint. Frequently the allowance is 
made too great or the shrinkage is not as much as anticipated, 
and it becomes necessary to cut off the top of the bank. On 
the other hand, the expense of raising the track after the road 
is in operation and the inevitable loss of ballast is so great 
that the danger of being required to fill up a sag should be 
avoided if possible. 



Fig. 60. 

A sharp and clear distinction should be made between the 
coefficient of extra height of aai embankment and the coefficient 
of shrinkage which determines how many cubic yards of settled 
embankment may be made from a definite volume of earth or 
rock measured in the excavation. The values quoted above 
for the Mississippi levees (from 10% to 25%) refers usually 
to a very soft soil and includes the effects other than actual 
contraction of volume. From 8% to 15% is usually quoted as 
the required extra height of embankments, although it is 
strenuously claimed by many that 3% or 2% is sufficient, 
or even that no allowance should be made. 

The coefficients to determine the amount of settled embank- 
ment which may be made from a given volume of earth or 
rock measured in the excavation, are necessarily subject to 
variation on account of the method employed and the amount 
of compression and settlement which will take place during 
the progress of the work. The following figures have the 
weight of considerable authority but, if in error, the coefficients 
are probably high rather than low: - ^ 



118 



RAILROAD CONSTRUCTION. 



98. 



Gravel or sand about 8% 

Clay '' 10% 

Loam. '' 12% 

Loose vegetable surface soil * ' 15% 

It may be noticed from the above table that the harder and 
cleaner the material the less is the contraction. Perfectly clean 
gravel or sand would not probably change volume appreciably. 
The above coefficients of shrinkage and expansion may be used 
to form the foUowino; convenient table: 



Material. 


To make 1000 cubic 

yards of embankment 

will require 


1000 cubic yards 
measured in exca- 
vation will make 


Gravel or sand 


1087 cubic yards 

1111 " 

1136 '* 

1176 " 

714 •* 

625 ;' 

measured in excavation 


920 cubic yarda 


Clay 


900 ** 


Loam. .... 


880 ** •• 


Loose vegetable soil 

Rock, large pieces 

' * small * ' 


850 ** 
1400 •• 
1600 *• 
of embankment. 



Since writing the above the following values have been 
adopted by the American Railway Engineering and Mainte- 
nance of Way Association as representing standard practice: 

Coefficients of Shrinkage Allowance for Depositinq 
Earthwork. 





Trestle filling. 


Raising under traffic. 


Black dirt 


15% 

10% 

6% 


il 


Clay 


Sand 


5% 







98. Methods of forming embankments. Embankments of 
moderate height are sometimes formed by scraping material 
with drag scrapers from ditches at the sides, especially if there 
is little or no cutting to be done in the immediate vicinity. 
Over a low level swampy stretch this method has the double 
advantage of building an embankment which is well above 
the general level and also provides generous drainage ditches 
which keep the embankment dry. Wheeled scrapers may be 
used economically up to a distance of 400 feet to excavate 



§98. EARTHWORK. . 119 

cuts and deposit the material on low embankments. Such 
methods have the advantage of compacting the embankments 
during construction and reducing future shrinkage. 

When carts are used, an embankment of any height may be 
formed by '' dumping over the end'' and building to the full 
height (or even higher to allow for shrinkage) as the embank- 
ment proceeds. The method is especially applicable when the 
material comes from a place as high as or higher than the 
grade-line, so that no up-hill hauling is necessary. Only a 
small contractor's plant is required for all of these methods. 

Trestles capable of carrying carts, or even cars and loco- 
motives, from which excavated material may be dropped, are 
found to be economical in spite of the fact that their cost is a 
construction expense. There is the disadvantage that such 
embankments require a long time to settle, but there are the 
advantages that the earth may be hauled by the train load 
from a distance of perhaps several miles, dumped from the 




Fig. 61. 



cars by train ploughs, or automatically dumped when the 
material is carried in patent dumping-cars, and all at a com- 
paratively small, cost per cubic yard. The disadvantages of 
slow settlement may be obviated, although at some additional 
cost, by making the trestle sufficiently strong to support regular 
traffic until the settlement is complete. 

During recent years cableways have been utilized to fill 
comparatively narrow but deep ravines from material obtain- 
able on either side of the ravine. This method obviates the 
construction of an excessively high trestle which might other- 
wise be considered necessary. 

Wheii an embankment is to be placed on a steep side hill 
which has a slippery clay surface, the embankment will some- 



120 RAILROAD CONSTRUCTION. §98. 

times slide down the hill, unless means are taken to prevent it. 
Some sort of bond between the old surface and the new material 
becomes necessary. This has sometimes been provided by 
cutting out steps somewhat as is illustrated in Fig. 61. It is 
possible that a deep ploughing of the surface would accom- 
plish the result just as effectively and much cheaper. The 
tendency to slip is generally due not only to the nature of the 
soil but also to the usual accompanying characteristic that the 
soil is wet and springy. The sub-surface drainage of such a 
place with tile drains will still further prevent such slipping, 
which often proves very troublesome and costly. 

COMPUTATION OF HAUL. 

99. Nature of subject. As will be shown later when analyz- 
ing the cost of earthwork, the most variable item of cost is that 
depending on the distance hauled. As it is manifestly imprac- 
ticable to calculate the exact distance to which every individual 
cartload of earth has been moved, it becomes necessary to devise 
a means which will give at least an equivalent of the haulage of 
all the earth moved. Evidently the average haul for any mass 
of earth moved is equal to the distance from the center of grav- 
ity of the excavation to the center of gravity of the embank- 
ment formed by the excavated material. As a rough approxi- 
mation the center of gravity of a cut (or fill) may sometimes be 
considered to coincide with the center of gravity of that part of 
the profile representing it, but the error is frequently very large. 
The center of gravity may be determined by various methods, 
but the method of the ''mass diagram" accomplishes the same 
ultimate purpose (the determination of the haul) with all-suffi- 
cient accuracy and also furnishes other valuable information, 

100. Mass diagram. In Fig. 62 let A'i^' . . , G' represent 
a profile and grade line drawn to the usual scales. Assume A' 
to be a point past which no earthwork will be hauled. Such 
a point is determined by natural conditions, as, for example, a 
river crossing, or one end of a long level stretch along which 
no grading is to be done except the formation of a low embank- 
ment from the material excavated from ample drainage ditches 
on each side. Above the profile draw an indefinite horizontal line 
{ACn in Fig, 62), which may be called the ''zero line." Above 
every station point in the profile draw an ordinate (above or be- 



§ 100. 



EARTHWORK. 



121 



low the zero line) which will represent the algebraic sum of 
the cubic yards of cub and fill 
(calling cut + and fill — ) from 
the point A^ to the point con- 
sidered. The computations of 
these ordinates should first be 
made in tabular form as shown 
below. In doing this shrinkage 
must be allowed for by consider- 
ing how much embankment 
would actually be made by so 
many cubic yards of excavation 
of such material. For example, 
it will be found that 1000 cubic 
yards of sand or gravel, measured 
in place (see § 97), will make 
about 920 cubic yards of embank- 
ment; therefore all cuttings in 
sand or gravel should be dis- 
counted in about this propor- 
tion. Excavations in rock should 
be increased in the proper 
ratio. In short, all excavations 
should be valued according to the 
amount of settled embankment 
that could be made from them. 
Place in the first column a list 
of the stations; in the second 
column,the number of cubicyards 
of cut or fill between each station 
and the preceding station; in 
the third and fourth columns, the kind of material and the proper 
shrinkage factor; in the fifth column, a repetition of the quan- 
tities in cubic yards, except that the excavations are diminished 
(or increased, in the case of rock) to the number of cubic yards 
of settled embankment which may be made from them. In 
the sixth column place the algebraic sum of the quantities in the 
fifth column (calling cuts + and fills — ) from the starting- 
point to the station considered. These algebraic sums at each 
station will be the ordinates, drawn to some scale, of the mass 
curve. The scale to be used will depend somewhat on whether 




122 



EAILEOAD CONSTRUCTION. 



§101. 



the work is heavy or light, but for ordinary cases a scale of 
5000 cubic yards per inch may be used. Drawing these ordi- 
nates to scale, a curve A, B, . . . G may be obtained by joining 
the extremities of the ordinates. 



Sta. 


Yards] -fl 


Material. 


Shrinkage 
factor. 


Yards, 

reduced 

for 

shrinkage. 


Ordinate 
in mass 
curve. 


46 + 70 













47 
48 

+ 60 
49 


4- 195 
4- 1792 
+ 614 

- 143 

- 906 

- 1985 

- 1721 

- 112 
4- 177 
4- 180 

- 52 

- 71 
+ 276 
+ 1242 
+ 1302 


Clayey soil 


- 10 per cent 

- 10 

- 10 


+ 175 
+ 1613 
+ 553 

- 143 

- 906 

- 1985 

- 1721 

- 112 
+ 283 
4- 289 

- 52 

- 71 
+ 249 
+ 1118 
+ 1172 


4- 175 
4- 1788 
+ 2341 
4 2198 


60 






+ 1292 


51 






- 693 


52 






- 2414 


4- 30 






- 2526 


53 

+ 70 
54 




Hard rock 


4- 60 per cent 
+ 60 


- 2243 

- 1954 

- 2006 


4- 42 






- 2077 


55 
56 
57 


Clayey soil 


— 10 per cent 

- 10 
~ 10 


- 1828 

- 710 
+ 462 



loi. Properties of the mass curve. 

1. The curve will be rising while over cuts and falling while 
over fills. 

2. A tangent to the curve will be horizontal (as at 5, D, E, 
Fj and G) when passing from cut to fill or from fill to cut. 

3- When the curve is helow the ''zero line'^ it shows that 
material must be drawn backward (to the left) ; and vice versa, 
when the curve is above the zero line it shows that material 
must be drawn forward (to the right) . 

4. When the curve crosses the zero line (as at A and C) it 
shows (in this instance) that the cut between A' and B^ will just 
provide the material required for the fill between B^ and €', and 
that no material should be hauled past C% or, in general, past 
any intersection of the mass curve and the zero line. 

5. If any horizontal line be drawn (as ab), it indicates that 
the cut and fill between a' and &' will just balance. 

6. When the center of gravity of a given volume of material 
is to be moved a given distance, it makes no. difference (at least 
theoretically) how far each individual load may be hauled or 
how any individual load may be disposed of. The summation 



f 



§ 101. EARTHWORK. 123 

of the products of each load times the distance hauled will be a 
constant, whatever the method, and will equal the total volume 
times the' movement of the center of gravity. The average 
haul, which is the movement of the center of gravity, will there- 
fore equal the summation of these products divided by the total 
volume. If we draw two horizontal parallel lines at an infini- 
tesimal distance dx apart, as at ah, the small increment of cut 
dx Sit a' w^ill fill the corresponding increment of fill at h' , and 
this material must be hauled the distance ab. Therefore the 
product of ah and dx, which is the product of distance -times 
volume, is represented by the area of the infinitesimal rectangle 
at ab, and the total area ABC represents the summation of 
volume times distance for all the earth movement between A' 
and C . This summation of products divided by the total 
volume gives the average haul. 

7. The horizontal line, tangent at E and cutting the curve 
at 6, /, and g, shows that the cut and fill between e' and E' will 
just balance, and that a possible method of hauling (whether 
desirable or not) would be to ^'borrow'' earth for the fill between 
C and e', use the material between D^ and £" for the fill between 
e' and D', and similarly balance cut and fill between E^ and /' 
and also between /' and g\ 

8. Similarly the horizontal line hklm may be drawn cutting 
the curve, which wnll show^another possible method of hauling. 
According to this plan, the fill between C and h^ would be 
made by borrowing; the cut and fill between h^ and A;' would 
balance; also that between A;' and V and between V and m\ 
Since the area ehDkE represents the measure of haul for the 
earth between e' and E^, and the other areas measure the corre- 
sponding hauls similarly, it is evident that the sum of the areas 
ehDkE and ElFmf, which is the measure of haul of all the 
material between e' and f, is largely in excess of the sum of 
the areas hDk, kEl, and IFm, plus the somewhat uncertain 
measures of haul due to borrowing material for e^h' and wasting 
the material between m' and /'. Therefore to make the meas- 
ure of haul a minimum a line should be drawn w^hich will make 
the sum of the areas betw^een it and the mass curve a minimum. 
Of course this is not necessarily the cheapest plan, as it implies 
more or less borrowing and wasting of material, which may 
cost more than the amoimt saved in haul. The comparison of 
the two methods is quite simple, however. iSirce the amount 



124 EAILROAD CONSTRUCTION. § 102. 

of fill between e' and h^ is represented by the difference of the 
ordinates at e and h, and similarly for m' and /', it follows that 
the amount to be borrowed between e' and W will exactly equal 
the amount wasted between m' and /'. By the first of the above 
methods the haul is excessive, but is definitely known from the 
mass diagram, and all of the material is utilized ; by the second 
method the haul is reduced to about one-half, but there is a 
known quantity in cubic yards wasted at one place and the same 
quantity borrowed at another. The length of haul necessary 
for the borrowed material would need to be ascertained; also 
the haul necessary to waste the other material at a place where 
it would be unobjectionable. Frequently this is best done by 
widening an embankment beyond its necessary width. The 
computation of the relative cost of the above methods will be 
discussed later (§ 116). 

9. Suppose that it were deemed best, after drawing the mass 
curve, to introduce a trestle between s' and v' , thus saving an 
amount in fill equal to tv. If such had been the original design, 
the mass curve would have been a straight horizontal line between 
s and t and would continue as a curve which would be at all 
points a distance tv above the curve vFmzfGg. If the line Ef is 
to be used as a zero line, its intersection with the new curve at x 
will show that the material between E^ and 2' will just balance 
if the trestle is used, and that the amount of haul will be meas- 
ured by the area between the line Ex and the broken line Estx. 
The same computed result may be obtained without drawing 
the auxiliary curve txn ... by drawing the horizontal line zy 
at a distance xz(=tv) below Ex. The amount of the haul can 
then be obtained by adding the triangular area between Es and 
the horizontal line Ex, the rectangle between st and Ex, and the 
irregular area between vFz and y . . . z (which last is evidently 
equal to the area between tx and E . . . x). The disposal of the 
material at the right of z^ would then be governed by the indica- 
tions of the profile and mass diagram which would be found at 
the right of g\ In fact it is difficult to decide with the best of 
judgment as to the proper disposal of material without having 
a mass diagram extending to a considerable distance each side 
of that part of the road under immediate consideration. 

102. Area of the mass curve. The area may be computed 
most readily by means of a planimeter, which is capable with 
reasonable care of measuring such areas with as great accuracy 



§ 103. EARTHWORK. 125 

as is necessary for this work. If no such instrument is obtain- 
able, the area may be obtained by an apphcatipn of " Simpson's 
rule." The ordinates will usually be spaced 100 feet apart. 
Select an even number of such spaces, leaving, if necessary, one 
or more triangles or trapezoids at the ends for separate and 
independent computation. Let ^/o . . . Vn be the ordinates, i.e., 
the munber of cubic yards at each station of the mass curve, or 
the figures of "column six" referred to in § 100. Let the uni- 
form distance between ordinates (^100 feet) be called 1, i.e., 
one station. Then the units of the resulting area will be cubic 
yards hauled one station. Then the 

Area = i[2/o + 4(2/1 + 2/3-^ • • .2/(n-l) + 2(2/2 + 2/4+ • • -2/(71-2) +2/ J- (70) 

When an ordinate occurs at a substation, the best plan is to 
ignore it at first and calculate the area as above. Then, if the 
difference involved is too great to be neglected, calculate the 
area of the triangle having the extremity of the ordinate at the 
substation as an apex, and the extremities of the ordinates at the 
adjacent stations as the ends of the base. This may be done by 
finding the ordinate at the substation that would be a propor- 
tional between the ordinates at the adjacent full stations. Sub- 
tract this from the real ordinate (or vice versa) and multiply the 
difference by |XT. An inspection will often show that the 
correction thus obtained would be too small to be worthy of con- 
sideration. If there is more than one substation between two 
full stations, the corrective area will consist of two triangles and 
one or more trapezoids which may be similarly computed, if 
necessary. 

When the zero line (Fig. 62) is shifted to eE, the drop from 
AC (produced) to E is known in the same units, cubic yards. 
This constant may be subtracted from the numbers ("column 
6," § 100) representing the ordinates, and will thus give, with- 
out any scaling from the diagram, the exact value of the modi- 
fied ordinates. 

103. Value of the mass diagram. The great value of the mass 
diagram lies in the readiness with which different plans for the 
disposal of material may be examined and compared. When 
the mass curve is once drawn, it will generally require only a 
shifting of the horizontal line to show the disposal of the material 
by any proposed method. The mass diagram also shows the 



126 RAILROAD CONSTRUCTION. § 104. 

extreme length of haul that will be required by any proposed 
method of disposal of material. This brings into consideration 
the ''limit of profitable haul/' which will be fully discussed in 
§ 116. For the present it may be said that with each method 
of carrying material there is some limit beyond which the expense 
of hauling will exceed the loss resulting from borrowing and 
wasting. With wheelbarrows and scrapers the limit of profit- 
able haul is comparatively short, with carts and tram-cars it is 
much longer, while with locomotives and cars it may be several 
miles. If, in Fig. 62, eE or Ef exceeds the limit of profitable 
haul, it shows at once that some such line as hklm should be 
drawn and the material disposed of accordingly. 

104. Changing the grade line. The formation of the mass 
curve and the resulting plans as to the disposal of material are 
based on the mutual relations of the grade line and the surface 
profile and the amounts of cut and fill which are thereby im- 
. plied. If the grade line is altered, every cross-section is altered, 
the amount of cut and fill is altered, and the mass curve is also 
changed. At the farther limit of the actual change of the grade 
line the revised mass curve will have (in general) a different 
ordinate from the previous ordinate at that point. From that 
point on, the revised mass curve will be parallel to its former 
position, and the revised curve may be treated similarly to the 
case previously mentioned in which a trestle was introduced. 
Since it involves tedious calculations to determine accurately 
how much the volume of earthwork is altered by a change in 
grade line, especially through irregular country, the effect on 
the mass curve of a change in the grade line cannot therefore 
be readily determined except in an approximate way. Raising 
the grade line will evidently increase the fills and diminish the 
cuts, and vice versa^ Therefore if the mass curve indicated, for 
example, either an excessively long haul or the necessity for 
borrowing material (implying a fill) and wasting material 
farther on (implying a cut), it w^ould be possible to diminish the 
fill (and hence the amount of material to be borrowed) by lower- 
ing the grade line near that place, and diminish the cut (and 
hence the amount of material to be wasted) by raising th( 
grade line at or near the place farther on. Whether the advan- 
tage thus gained would compensate for the possibly injurious 
effect of these changes on the grade line would require patient 
investigation. But the method outlined shows how the mass 



)r 

I 



§ 105 



EARTHWORK. 



127 



curve might be used to indicate a possible change in grade hne 
which might be demonstrated to be profitable. 

105. Limit of free haul. It is sometimes specified in con- 
tracts for earthwork that all material shall be entitled to free 
haul up to some specified limit, say 500 or 1000 feet, and that 
all material drawn farther than that shall be entitled to an 
allowance on the excess of distance. It is manifestly imprac- 
ticable to measure the excess for each load, as much so as to 
measure the actual haul of each load. The mass diagram also 
solves this problem very readily. Let Fig. 63 represent a pro- 




FiG. 63. 



file and mass diagram of about 2000 feet of road, and suppose 
that 800 feet is taken as the limit of free haul. Find two points, 
a and h, in the mass curve which are on the same horizontal line 
and which are 800 feet apart. Project these points down to a' 
and h\ Then the cut and fill between a^ and V will just balance, 
and the cut between A ' and a' will be needed for the fill between 
V and C\ In the mass curve, the area between the horizontal 
line ah and the curve aBh represents the haulage of the material 
between a' and h' , which is all free. The rectangle dbmn repre- 
sents the haulage of the material in the cut A^a^ across the 800 
feet from a' to h' . This is also free. The sum of the two areas 
A am and hnC represents the haulage entitled to an allowance, 
since it is the summation of the products of cubic yards times 
the excess of distance hauled. 

If the amount of cut and fill was symmetrical about the point 



128 RAILROAD CONSTRUCTION. § 106, 

B', the mass curve would be a symmetrical curve about the 
vertical line through B^ and the two limiting lines of free haul 
would be placed symmetrically about B and B' , In general 
there is no such symmetry, and frequently the difference is con- 
siderable The area aBbnm will be materially changed accord- 
ing as the two vertical hnes am and 6n, always 800 feet apart, 
are shifted to the right or left. It is easy to show that the area 
aBbnm is a maximum when ah is horizontal. The minimum 
value w^ould be obtained either w^hen m reached A or n reached 
C, depending on the exact form of the curve. Since the posi- 
tion for the minimum value is manifestly unfair, the best definite 
value obtainable is the maximum, which must be obtained as 
above described. Since aBbnm is made maximum, the remainder 
of the area, which is the allowance for overhaul, becomes a mini- 
mum. The areas Aam and bCn may be obtained as in § 102. 
If the whole area AaBbCA has been previously computed, it 
may be more convenient to compute the area aBbnm and sub- 
tract it from the total area. 

Since the intersections of the mass curve and the ''zero line'' 
mark limits past which no ma.terial is drawn, it follows that 
there will be no allowance for overhaul except where the dis- 
tance between consecutive intersections of the zero line and mass 
curve exceeds the limit of free haul. 

Frequently all allowances for overhaul are disregarded; the 
profiles, estimates of quantities, and the required disposal of 
material are shown to bidding contractors, and they must then 
make their own allowances and bid accordingly. This method 
has the advantage of avoiding possible disputes as to the amount 
of the overhaul allowance, and is popular with railroad com- 
panies on this account. On the other hand the facility with 
which different plans for the disposal of material may be studied 
and compared by the mass-curve method facilitates the adoption 
of the most economical plan, and the elimination of uncertainty 
will frequently lead to a safe reduction of the bid, and so the 
method is valuable to both the railroad company and the con- 
tractor. 



ELEMENTS OF THE COST OF EARTHWORK. 



I 



io6. Analysis of the total cost into items. The variation in 
the total cost of excavating earthwork, hauling it a greater ori 
less distance, and forming with it an embankment of definitel 



cer ori 

•J 



§ 107. EARTHWORK. 129 

form or wasting it on a spoil bank, is so great that the only 
possible method of estimating the cost under certain assumed 
conditions is to separate the total cost into elementary items. 
Ellwood Morris was perhaps the first to develop such a method 
— see Journal of the Franklin Institute, September and October, 
1841. Trautwine used the same general method with some 
modifications. The following analysis will follow the same 
general plan, will quote some of the figures given by Morris 
and by Trautwine, but will also include facts and figures better 
adapted to modern conditions. Since every item of cost (except 
interest on cost of plant and its depreciation) is a direct function 
of the current price of common labor, all calculations will be 
based on the simple unit of $1 per day. Then the actual cost 
may be obtained by multiplying the calculated cost under the 
given conditions by the current price of day labor. "WTien 
possible, figures will be quoted giving the cost of all items of 
work on a loose sandy soil which is the easiest to work and also 
for the cost of the heaviest soils, such as stiff clay and hard pan. 
These represent the extremes, excluding rock, which will be 
treated separately. The cost of intermediate grades may be 
interpolated between the extreme values according to the 
judgment of the engineer as to the character of the soil. 

The possible division into items varies greatly according to 
the method adopted, but the differentiation into items given 
below (which is strictly applicable to the old fashioned simpler 
methods of work) can usually be applied to any other method 
by merely combining or eliminating some of the items. The 
items are 

1. Loosening the natural soil. 

2. Loading the soil into whatever carrier may be used. 

3. HauHng excavated material from excavation to embank- 

ment or spoil bank. 

4. Spreading or distributing the soil on the embankment. 

5. Keeping roadways or tracks in good running order. 

6. Trimming cuts to their proper cross-section (sometimes 

called ' ' sandpapering ' ') . 

7. Repairs, wear, depreciation, and interest on cost of plant. 

8. Superintendence and incidentals. 

107. Item I. Loosening, (a) Ploughs. Very light sandy 
soils can frequently be shovelled without any previous loosen- 
ing, but it is generally economical, even with very light material, 



130 RAILROAD CONSTRUCTION. § 107. 

to use a plough. Morris quotes, as the results of experiments, 
that a three-horse plough would loosen from 250 to 800 cubic 
yards of earth per day, which at a valuation of $5 per day 
would make the cost per yard vary from 2 cents to 0.6 cent. 
Trautwine estimates the cost on the basis of two men handling 
a two-horse plough at a total cost of $3.87 per day, being $1 
each for the men, 75 c. for each horse, and an allowance of 
37 c. for the plough, harness, etc. From 200 to 600 cubic yards - 
is estimated as a fair day's work, which makes a cost of 1.9 c. ■I 
to 0.65 c. per yard, which is substantially the same estimate 
as above. Extremely heavy soils have sometimes been loosened 
by means of special ploughs operated by traction-engines. 

Gillette estimates that "8b two-horse team with a driver and - 
a man holding the plough will loosen 25 cubic yards of fairly I! 
tough clay, or 35 cubic yards of gravel and loam per hour." 
For ten hours per day this would be 250 to 350 cubic yards 
per day. These values are neither as high nor as low as the 
extremes above noted. It is probably very seldom that a soil 
will be so light that a two-horse (or three-horse) plough can 
loosen as much as 600 (or 800) cubic yards per day. 

It is sometimes necessary to plough up a macadamized street. 
This may be done by using as a plough a pointed steel bar 
which is fastened to a very strong plough frame. A prelimi- 
nary hole must be made which will start the bar under the 
macadam shell. Then, as the plough is drawn ahead, the shell 
is ripped up. Four or six horses, or even a traction-engine, 
are used for such work. Gillette quotes two such cases where 
the cost of such loosening was 2 c. and 6 c. per cubic yard, 
with common labor at 15 c. per hour. Two-thirds of such 
figures w411 reduce them to the $1 per day basis. The cost for 
ploughing on the $1 per day basis may therefore be summarized 
as follows: 



For very loose sandy soils 0.6c. per cubic yard 

*' '' heavy clay '' 2.0 c. '' '' '' 

'* hard pan and macadam, up to .. . 4.0 c. '/ ** " 

(b) Picks. When picks are used for loosening the earth, as 
is frequently necessary and as is often done when ploughing 
would perhaps be really cheaper, an estimate * for a fair day's 



* Trautwine. 



I 



ji 



§ 108. EARTHWORK. 131 

work is from 14 to 60 cubic yards, the 14 yards being the esti- 
mate for stiff clay or cemented gravel, and the 60 yards the esti- 
mate for the lightest soil that would require loosening. At $1 
per day this means about 7 c. to 1.7 c. per cubic yard, which is 
about three times the cost of ploughing. Five feet of the face 
is estimated * as the least width along the face of a bank that 
should be allowed to enable each laborer to work with freedom 
and hence economically. 

(c) Blasting. Although some of the softer shaly rocks may 
be loosened with a pick for about 15 to 20 c. per yard, yet rock 
in general, frozen earth, and sometimes even compact clay are 
most economically loosened by blasting. The subject of blast- 
ing will be taken up later, §§117-123. 

(d) Steam-shovels. The items of loosening and loading 
merge together with this method, which will therefore be treated 
in the next section. 

io8. Item 2. Loading, (a) Hand-shovelling. Much depends 
on proper management, so that the shovellers need not wait un- 
duly either for material or carts. With the best of management 
considerable time is thus lost, and yet the intervals of rest 
need not be considered as entirely lost, as it enables the men to 
work, while actually loading, at a rate which it would be physi- 
cally impossible for them tcr maintain for ten hours. Seven 
shovellers are sometimes allowed for each cart; otherwise there 
should be five, two on each side and one in the rear. Economy 
requires that the number of loads per cart per day should be 
made as large as possible, and it is therefore wise to employ as 
mary shovellers as can work without mutual interference and 
without wasting time in waiting for material or carts. The 
figures obtainable for the cost of this item are unsatisfactory on 
account of their large disagreements. The following are quoted 
as the number of cubic yards that can be loaded into a cart by 
an average laborer in a working day of ten hours, the lower 
estimate referring to heavy soils, and the higher to light sandy 
soils: 10 to 14 cubic yards (Morris), 12 to 17 cubic yards (Has- 
koll), 18 to 22 cubic yards (Hurst), 17 to 24 cubic yards (Traut- 
wine), 16 to 48 cubic yards (Ancelin). As these estimates are 
generally claimed to be based on actual experience, the discrep- 
ancies are probably^ due to differences of management. If the 

'" * Hurst. 



132 RAILROAD CONSTRUCTION. § 108. 

average of 15 to 25 cubic yards be accepted, it means, on the 
basis of $1 per day, 6.7 c. to 4 c, per cubic yard. These esti- 
mates apply only to earth. Rockwork costs more, not only 
because it is harder to handle, but because a cubic yard of solid 
rock, measured in place, occupies about 1.8 cubic yards when 
broken up, while a cubic yard of earth will occupy about 1.2 
cubic yards. Rockwork will therefore require about 50% more 
loads to haul a given volume, measured in place, than will the 
same nominal volume of earthwork. The above authorities give 
estimates for loading rock varying from 6.9 c. to 10 c. per cubic 
yard. The above estimates apply only to the loading of carts 
or cars with shovels or by hand (loading masses of rock). The 
cost of loading wheelbarrows and the cost of scraper work will 
be treated under the item of hauling. 

(b) Steam-shovels.* Whenever the magnitude of the work 
will warrant it there is great economy in the use of steam-shovels. 
These have a ^^ bucket" or ^^ dipper" on the end of a long beam, 
the bucket having a capacity varying from J to 2J cubic yards. 
Steam-shovels handle all kinds of material from the softest 
earth, to shale rock, earthy material containing large boulders, 
tree-stumps, etc. The record of work done varies from 200 to 
1000 cubic yards in 10 hours. They perform all the work of 
loosening and loading. Their economical working requires that 
the material shall be hauled away as fast as it can be loaded, 
which usually means that cars on a track, hauled by horses or 
mules, or still better by a locomotive, shall be used. The ex- 
penses for a steam'^shovel, costing about S5000, will average 
about $1000 per month. Of this the engineer may get $100; the 
fireman $50 ; the cranesman $90 ; repairs perhaps $250 to $300 ; 
coal, from 15 to 25 tons, cost very variable on account of expen- 
sive hauling; water, a very uncertain amount, sometimes costing 
$100 per month; about five laborers and a foreman, the laborers 
getting $1.25 per day and the foreman $2.50 per day, which will 
amount to $227.50 per month. This gang of laborers is employed 
in shifting the shovel when necessary, taking up and relaying 

* For a thorough treatment of the capabilities, cost, and management 
of steam-snovels the reader is referred to " Steam-shovels and Steam-shovel 
Work," by E. A. Hermann. D. Van Nostrand Co., New York. 

This book is now out of print. " Earthwork and its Cost," by H. P. Gil- 
lette, to which the student is referred for a more elaborate exposition of the 
subject, has used many of Hermann's cuts. 



§ 108. EARTHWORK. , 133 

tracks for the cars, shifting loaded and unloaded cars^ etc. In 
shovelling through a deep cut^ the shovel is operated so as to 
undermine the upper parts of the cut* which then fall down 
within reach of the shovel, thus increasing the amount of material 
handled for each new position of the shovel. If the material is 
too tough to fail down by its own weight, it is sometimes found 
economical to employ a gang of men to loosen it or even blast it 
rather than shift the shovel so frequently. Non-condensing 
engines of 50 horse-power use so much water that the cost of 
water-supply becomes a serious matter if water is not readily 
obtainable. The lack of water facilities will often justify the 
construction of a pipe line from some distant source and the 
installation of a steam-pump. Hence the seemingly large 
estimate of $100 per month for water-supply, although under 
favorable circumstances the cost may almost vanish. The larger 
steam-shovels will consume nearly a ton of coal per day of 10 
hours. The expense of hauling this coal from the nearest rail- 
road or canal to the location of the cut is often a very serious 
item of expense and may easily double the cost per ton. Some 
steam-shovels have been constructed to be operated by electricity 
obtained from a plant perhaps several miles away. Such a 
method is especially advantageous when fuel and water are diffi- 
cult to obtain. ^ 

The following general requirements and specifications were 
recommended in 1907 by the American Hallway Engineering 
and Maintenance of Way Association: 

Three important cardinal points should be given careful 
attention in the selection of a steam-shovel. These are in their 
order 

(1) Care in the selection, inspection and acceptance of all 
material that enters into every part of the machine. 

(2) Design for strength. 

(3) Design for production. 

GENERAL SPECIFICATIONS. 

Weight of shovel: Seventy (70) tons. 
Capacity of dipper: Two and one-half (2 J) yards. 
Steam pressure: One hundred and twenty (120) pounds. 
Clear height above rail of shovel track at which dipper should 
unload: Sixteen (16) feet. 



134 RAILROAD CONSTRUCTION. § 109. 

Depth below rail of shovel track at which dipper should dig 
Four (4) feet. 

Number of movements of dipper per minute from time of 
entering banli to entering bank: Three (3). 

Character of hoist: Cable. 

Character of swing: Cable. 

Character of housing: Permanent for all employes. 

Capacity of tank: Two thousand (2000) gallons. 

Capacity of coal-bunker: Four (4) tons. 

Spread of jack arm: Eighteen (18) feet. A special short arm 
should be provided. 

Form of steam-shovel track: "T'' rails on ties. 

Length of rails for ordinary work: Six (6) feet. 

Form of rail joint: Strap. 

Manufacturers of steam-shovels will eometimes "guarantee" 
that certain of their shovels will excavate, say 3000 cubic yards 
of earth per day of ten hours. Even if it were possible for a 
shovel to fill a car at the rate of 5 cubic yards per minute, it is 
always impracticable to maintain such a speed, since a shovel 
must always wait for the shifting of cars and for the frequent 
shifting of the shovel itself. There are also delays due to 
adjustments and minor breakdowns. The best shovel records 
are made when the cars are large — other things being equal. 
The item of interest and depreciation of the plant is very large 
in steam-shovel work. This will be discussed further later. 
The cost of loading alone will usually come to between 3 and 
4 c. per cubic yard. The cost of shifting the cars so as to 
place them successively under the shovel, haul them to the 
dumping place, dump them and haul them back, will generally 
be as much more. Gillette quotes five jobs on one railroad 
where the total cost for loading and hauling varied from 5.9 c. 
to 11.4 c. per cubic yard. But as these figures are based on 
car measurement, the cost per cubic yard in flace measure-a 
ment must be increased about one-fourth, or from 7.4 c. tdl 
14.2 c. 

109. Item 3. Hauling. The cost of hauling depends on 
the number of round trips per day that can be made by each 
vehicle employed. As the cost of each vehicle is practically the 
same whether it makes many trips or few, it becomes important 
that the number of trips should be made a maximum, and to that 
end there should be as little delay as possible in loading and 



-li 



§ 109 EARTHWOKK. 135 

unloading. Therefore devices for facilitating the passage of the 
vehicles have a real money value. 

(a) Carts. The average speed of a horse hauling a two- 
wheeled cart has been found to be 200 feet per minute, a little 
slower when hauling a load and a little faster when returning 
empty. This figure has been repeatedly verified. It means an 
allowance of one minute for each 100 feet (or '^station*') of 
*4ead — the lead being the distance the earth is hauled." The 
time lost in loading, dumping, waiting to load, etc., has been 
foimd to average 4 minutes per load. Representing the num- 
ber of Nations (100 feet) of lead by s, the number of loads 
handled in 10 hours (600 minutes) would be 600 ^(s + 4). The 
number of loads per cubic yard, measured in the bank, is differ- 
entiated by Morris into three classes, viz. : 

3 loads per cubic yard in descending hauling; 
Si " '' " '' '' level hauling; and 
4. '' '' '' '' '' ascending hauling. 

Attempts have been made to estimate the effect of the grade 
of the roadway by a theoretical consideration of its rate, and of 
the comparative strength of a horse on a level and on various 
grades. While such computations are always practicable on a 
railway (even on a temporary construction track), the traction 
on a temporary earth roadway is alw^ays very large and so very 
variable that any refinements are useless. On railroad earth- 
work the hauling is generally nearly level or it is descending — 
forming embankments on low ground with material from cuts in 
high ground. The only common exception occurs when an 
embankment is formed from borrow-pits on low ground. One 
method of allow^ing for ascending grade is to add to the hori- 
zontal distance 14 times the difference of elevation for work 
with carts and 24 times the difference of elevation for work 
with wheelbarrows, and use that as the lead. For example, 
using carts, if the lead is 300 feet and there is a difference of 
elevation of 20 feet, the lead would be considered equivalent to 
300 + (14X20) =580 feet on a level. 

Trautwine assumes the average load for all classes of work 
t© be \ cubic yard, which figure is justified by large experience. 
Using one figure for all classes of work simplifies the calculations 
and gives the niunber of cubic yards carried per day of 10 hours 

equal to , . Dividing the cost of a cart per day by the 



136 RAILROAD CONSTRUCTION. § 109. 

number of cubic yards carried gives tbe cost of hauling per 
yard. In computing the cost of a cart per day, Trautwine 
refers to the practice of having one driver manage four carts, 
thus making a charge of 25 c. per day for each cart for the driver. 
Although this might be an economical method when the haul is 
very long, it is not economical for short hauls. A safer estimate 
is to allow not more than two carts per driver and in many 
cases a driver for each cart. Some contractors employ a driver 
for each cart and then require that the drivers shall assist in 
loading. The policy to be adopted is sometimes dependent on 
labor union conditions, which may demand that drivers must 
not assist in loading. The supply of labor and the amount of 
work on hand have a great influence on the methods of work 
which a contractor may adopt, for a strike will often disarrange 
all plans. 

The cost of a horse and cart must practically include a 
charge for the time of the horse on Sundays, rainy days and 
holidays. The cost of repairs of cart and harness is generally 
included in this item for simplicity, but, under a strict applica- 
tion of the analysis suggested in § 106, it should properly be 
included under Item 7, Repairs, etc. 

Since the time required for loading loose rock is greater 
than for earthwork, less loads will be hauled per day. The time 
allowance for loading, etc., is estimated by Trautwine as 6 
minutes instead of 4 as for earth. Considering the great ex- 
pansion of rock when broken up (see § 97), one cubic yard of 
solid rock, measured in place, would furnish the equivalent of 
five loads of earthwork of J cubic yard. Therefore, on the 
basis of five loads per cubic yard, the number of cubic yards 

handled per day per cart would be -^7 — —^. 

o(s + b) 

Let C represent the daily cost of a horse and cart and of 

the proportional cost of the driver (according to the number of 

carts handled by one driver), then the cost per cubic yard, 

measured in the cut, for hauling may be given by the formula: 



Cost per cu. yd. of hauling earth in carts = ^r^ — - 

a (c u u u u „_^v <' <' _ CX5(s + 6) 



§ 109. EARTHWORK. 137 

(b) Wagons. For longer leads (i.e., from | to f of a mile) 
wagons drawn by two horses have been found most economical. 
The wagons have bottoms of loose thick narrow boards and are 
unloaded very easily and quickly by lifting the individual boards 
and breaking up the continuity of the bottom, thus depositing 
the load directly underneath the wagon. The capacity is about 
one cubic yard. The cost may be estimated on the same prin- 
ciple as that for carts. 

The number of wagon trips per 10 hours will depend some- 
what on the management of the shovellers. Too many shovel- 
lers per wagon is not economical, measured in yards shovelled 
per man, although it may reduce the time consumed in loading 
any one wagon. At an average figure of 20 cubic yards, 
measured in place, per shoveller per 10 hours, seven shovellers 
would load 14 cubic yards per hour or one cubic yard in 4.3 
minutes. This would be the allowance for a wagon with a 
capacity of about IJ yards of loose earth. Adding time for 
unloading, waiting to load and other possible '' lost time,'* there 
is probably a total of six minutes. This figure will vary very 
considerably according to the number of shovellers per wagon, 
the capacity of the wagon, the type of wagon (whether self- 
dumping) and other details in the method of management. 
Adopting six minutes as the time used for loading, unloading, 
and other ''lost time," the formula becomes. 

Cost per cubic yard of hauling in wagons = -^--r — -^ . . . (71a) 

in which C is the cost of the wagon, team and driver per day 
of 10 hours; s is the distance hauled in stations of 100 feet, 
and c is the capacity of the wagon in cubic yards, place meas- 
urement , which should be about three fourths of the nominal 
capacity of the wagon for earth and about sixty per cent when 
handling rock. 

(c) Wheelbarrows. Gillette has computed from observa- 
tions that a man will trundle a wheelbarrow at the rate of 250 
feet per minute or 1.25 stations of lead per minute for the round 
trip. The time required for loading is estimated at 2J minutes 
and for unloading, adjusting wheeling planks, short rests, etc., 
I minute, or a total of three minutes per trip for all purposes 
except hauling. Gillette allows for a load only 1/15 cubic yard, 



138 RAILROAD CONSTRUCTIONv § 109. 

measured in place, or about 1/11 yard, 2,5 cubic feet, on the 
wheelbarrow. With notation as before 

Cost per cubic yard of loading and \ _ CXl5(1.25s + 3) 
hauling earth in wheelbarrows / 600 * ^ 

In this equation C is the cost of both loading and hauling, and 
usually includes the allowance (Item 7) for the cost, repairs 
and depreciation of the wheelbarrows, whose service is very 
short lived. Trautwine estimates this at five cents per day or 
a total of $1.05 for labor and wheelbarrow. 

The number of wheelbarrow loads required for a cubic yard 
of rock, measured in place, is about twenty-four. The time 
required for loading should also be increased about one fourth; 
the time required for all purposes except hauling is therefore 
about 3.75 minutes, and the corresponding equation becomes 

Cost per cubic yard of loading and \ _ CX 24(1.255 + 375) ._ 
hauling rock in wheelbarrows / 600 * 

(d) Scrapers. These are made in three general ways, ''buck'* 
scrapers, *'drag" scrapers and ^'wheeled '' scrapers. The buck 
scraper in its original form consisted merely of a wide plank, 
shod with ani iron strap on the lower edge and provided with 
a pole and a small platform on which the driver may stand to 
weight it down. The earth is not loaded on to any receptacle 
and carried, but is merely pushed over the ground. Notwith- 
standing the apparent inefficiency of the method, its extreme « * 
simplicity has caused its occasional adoption for the construe- II 
tion of canal embankments out of material from the bed of the 
canal. The occasions are rare when their use for railroad work 
would be practicable, and even then drag scrapers would prob- 
ably be preferable. 

A drag scraper is an immense "scoop shovel" about three feet 
long and three feet wide. There are usually two handles and a - ! 
bail in front by which it is dragged by a team of horses. The ' 
nominal capacity varies from 7.5 cubic feet for the largest sizes, 
down to 3 cubic feet for the '^one-horse'' size, but these figures . 
must be discounted by perhaps 40 or 50% for the actual average 1 1 
volume (as measured in the cut) loaded on during one scoop. 
The expansion of the earth during loosening is alone respons- 



J 



§ 109. EARTHWORK. 139 

ible for a discount of 25%. These scrapers cost from $10 to 
$18. 

A wheeled scraper is essentially an extra-large drag scraper 
which may be raised by a lever and carried on a pair of large 
wheels. Their nominal capacity ranges from 10 to 17 cubic feet, 
which should usually be liberally discounted when estimating 
output. They are loaded by dropping the scoop so that it 
scrapes up its load. The lever raises the scoop so that the load 
is carried on wheels instead of being dragged. At the dump the 
scoop is tipped so as to unload it. The movement of the 
scraper is practically continuous. They cost from $40 to 
$75. Their advantages over drag scrapers consist (1) in their 
greater capacity, (2) in the economy of transporting the load 
on wheels instead of by dragging, and (3) in the far greater 
length of haul over w^hich the earth may be economically 
handled. 

Morris estimated the speed of drag scrapers to be 140 feet per 
minute, or 70 feet of lead per minute. The "lead" should be 
here interpreted as the average distance from the center of the 
pit to the center of the dump. Gillette declares the speed to be 
220 feet per minute. Some of this variation may be due to dif- 
ferences in the method of measuring the distance actually trav- 
elled, especially when the lead is very short, since the scraper 
teams must always travel a considerable extra distance at each 
end in order to turn around most easily. This extra distance is 
practically constant whether the lead is long or short. Gillette 
quotes an instance where the length of lead was actually about 
20 feet, but the scraper teams travelled about 150 feet for each 
load carried. On this a(?count Gillette adopts a minimum of 
75 feet of lead no matter how short the lead actually may be. 
Of course the speed depends considerably on how strictly the 
men are kept to their work and also on the care which may be 
taken to obtain a full load for each scraper. As a compromise 
between Morris's and Gillette's estimates we may adopt the con- 
venient rate of speed of 200 feet per minute, or 100 feet of lead 
per mmute. There should also be allowed for the time lost 
in loading and unloading and for travelling the extra distance 
travelled by the teams in making the circuit, If minutes. Allow- 
ing the average value of seven loads per cubic yard and letting 
C represent the cost of scraper team and driver per day, we 
tiave for the cost as follows: 



140 RAILROAD CONSTRUCTION. § 109 

CX7(s + l-i) 



Cost per cubic yard of loading and 
hauling earth in drag scrapers 



600 



(73) 



In this formula C should include the cost of not only the 
driver, team, and scraper, but also the proper proportion of 
the wages of an extra man, who assists each driver in loading 
his scraper, and whose wages should be divided among the two 
(or three) scrapers to which he is assigned. Scraper work 
nearly always implies ploughing, the cost of which should be 
computed as under Item 1. 

When a low embankment is formed from borrow-pits on each 
side of the road, it may be done with scrapers, which move fronx 
one borrow-pit to the other, taking a load alternately from each 
side to the center and making but one half turn for each load 
carried. This reduces the time lost in turning by one third of a 
minute and reduces the constant in the numerator in Eq. (73) 
from 1|^ to 1. In this case the lead will usually be not greater 
than 75 feet, and therefore, if we consider this as a minimum 
value, s will ordinarily equal .75 and the quantity in the paren- 
thesis will equal 1.75. 

When using wheeled scrapers the catalogue capacity, which 
varies from 9 or 10 feet for a No. 1 scraper to 16 or 17 feet for 
a No. 3 scraper, must be reduced to 5 loads per cubic yard 
(place measurement) for a No. 1 scraper and to 2^ loads per 
cubic yard for a No. 3, not only on account of the expansion of 
the earth during loosening, but also on account of the imprac- 
ticability of loading these scrapers to their maximum nominal 
capacity. When the haul or lead for wheeled scrapers is 300 
feet or over, it will be justifiable to employ shovellers to fill up 
the bowl of the shovel, especially when the soil is tough and 
when it is impracticable to fill the shovel even approximately 
full by the ordinary method. A snatch team to assist in load- 
ing the scrapers it also economical, especially with the larger 
scrapers. The proportionate number of snatch teams to the 
total number of scrapers of course depends on the length of 
haul. The cost of these extra shovellers and extra snatch teams 
must be divided proportionally among the number of scrapers 
assisted, in determining the value C in the formula given below. 
The extra time to be allowed on account of turning, loading, 
and dumping is about IJ minutes. The speed is considered 
one station of lead per minute as before. If we call C the average 



I 



§ 109. EARTHWORK. 141 

daily cost of one scraper and n the capacity of the scraper, or 
the number of loads per cubic yard, we may write the following 
formula: 

Cost per cubic yard of loading and 1 _ CXn(s + li) /Yq ^ 

hauling earth in wheeled scrapers J 600 

(e) Cars and horses. The items of cost by this method are 
(a) charge for horses employed, (b) charge for men employed 
strictly in hauling, (c) charge for shifting rails when necessary, 
(d) repairs, depreciation, and interest on cost of cars and track. 
Part of this cost should strictly be classified under items 5 and 
7, mentioned in § 106, but it is perhaps more convenient to 
estimate them as follows: 

The traction of a car on rails is so very small that grade 
resistance constitutes a very large part of the total resistance 
if the grade is 1% or more. For all ordinary grades it is 
sufficiently accurate to say that the grade resistance is to 
the gross weight as the rise is to the distance. If the distance 
is supposed to be measured along the slope, the proportion is 
strictly true; i.e., on a 1% grade the grade resistance is 1 lb. 
per 100 of weight or 20 lbs. per ton. If the resistance on a 
level at the usual velocity is -p|-Q, a grade of 1:120 (0.83%) will 
exactly double it. If the material is hauled down a grade of 
1:120, the cars will run by gravity after being started. The 
work of hauling will then consist practically of hauling the 
empty cars up the grade. The grade resistance depends only 
on the rate of grade and the weight, but the tractive resistance 
will be greater per ton of weight for the unloaded than for the 
loaded cars. The tractive power of a horse is less on a grade 
than on a level, not only because the horse raises his own weight 
in addition to the load, but is anatomically less capable of 
pulling on a grade than on a level. In general it will be pos- 
sible to plan the work so that loaded cars need not be hauled up 
a grade, unless an embankment is to be formed from a low 
borrow-pit, in which case another method would probably be 
advisable. These computations are chiefly utilized in design- 
ing the method of work — the proportion of horses to cars. An 
example may be quoted from English practice (Hurst), in which 
the cars had a capacity of 3^ cubic yards, weighing 30 cwt. 
empty. Two horses took five "wagons" | of a mile on a level 



142 KAILROAD CONSTRUCTION. § 109, 

railroad and made 15 journeys per day of 10 hours, i.e., they 
handled 250 yards per day. In addition to those on the 
"straight road," another horse was employed to make up 
the train of loaded wagons. With a short lead the straight- 
road horses were employed for this purpose. In the above 
example the number of men required to handle these cars, 
shift the tracks, etc., is not given, and so the exact cost of the 
above work cannot be analyzed. It may be noticed that the 
two horses travelled 22J miles per day, drawing in one direction 
a load, including the weight of the ears, of about 57,300 lbs., 
or 28.65 net tons. Allowing ^2^ ^^ the necessary tractive 
force, it would require a pull of 477.5 lbs., or 239 lbs. for each 
horse. With a velocity of 220 feet per minute this would amount 
to li horse-power per horse, exerted for only a short time, 
however, and allowing considerable time for rest and for draw- 
ing only the empty cars. Gillette claims that the rolling re- 
sistance for such ears on a contractor's track should be con- 
sidered as 40 lbs. per ton (the equivalent of a 2% grade) and 
quotes many figures to support the assertion. Unquestionably 
the resistance on tracks with very light rails, light ties with 
wide spacing and no tamping, would be very great and might 
readily amount to 40 lbs. per ton. In the above case, the 
resistance could not have been much if any over y-J -q. A re- 
sistance of 40 lbs. per ton would have required each horse to 
pull about 573 lbs. for nearly five hours per day, beside pulling 
the empty cars the rest of the time. This is far greater exertion 
than any ordinary horse can maintain. The cars generally used 
in this country have a capacity of 1 J cubic yards and cost about 
$65 apiece. Besides the shovellers and dumping-gang, several 
men and a foreman will be required to keep the track in order 
I and to make the constant shifts that are necessary. Two trains 
' are generally used, one of which is loaded while the other is run 
to the dump. Some passing-place is necessary, but this is 
generally provided by having a switch at the cut and running 
the trains on each track alternately. This insures a train of 
cars always at the cut to keep the shovellers employed. The 
cost of hauling per cubic yard can only be computed when the 
number of laborers, cars, and horses employed are known, and 
these will depend on the lead, on the character of the excavation, 
on the grade, if any, etc., and must be so proportioned that the 
shovellers need not wait for cars to fill, nor the dumping-gang 



§ 109. EARTHWORK. 143 

for material to handle, nor the horses and drivers for cars to 
haul. Much skill is necessary to keep a large force in smooth 
rimning order. 

(f) Cars and locomotives. 30-lb. rails are the lightest that 
should be used for this work, and 35- or 40-lb. rails are better. 
One or two narrow-gauge locomotives (depending on the length 
of haul), costing about $2500 each, will be necessary to handle 
two trains of about 15 cars each, the cars having a capacity of 
about 2 cubic yards and costing about $100 each. Some cars 
can be obtained as low as $70. A force of about five men and 
a foreman will be required to shift the tracks. The track- 
shifters, except the foreman, may be common laborers. The 
dumping-gang will require about seven men. Even when the 
material is all taken down grade the grades may be too steep for 
the safe hauling of loaded cars down the grade, or for hauling 
empty cars up the grade. Under such circumstances temporary 
trestles are necessary to reduce the grade. When these are 
used, the uprights and bracing are left in the embankment — • 
only the stringers being removed. This is largely a necessity, 
but is partially compensated by the fact that the trestle forms a 
core to the embankment which prevents lateral shifting during 
settlement. The average speed of the trains may be taken as 
10 miles per hour or 5 miles of lead per hour. The time lost 
in loading and unloading is estimated (Trautwine) as 9 minutes 
or .15 of an hour. The number of trips per day of 10 hoiu-s 

^.^ ^ ^^j 10 ^^ 50 , 

^ ^(miles of lead) + .15 (miles of lead) + .75* 

course this quotient must be a whole number. Knowing the 
number of trains and their capacity, the total number of cubic 
yards handled is known, which, divided into the total daily cost 
of the trains, will give the cost of hauling per yard. The daily 
cost of a train will include 

(a) Wages of engineer, who frequently fires his own engine; 

(6) Fuel, about J to 1 ton of bituminous coal, depending on 
work done ; 

(c) Water, a very variable item, frequently costing $3 to $5 
per day; 

(d) Repairs, variable, frequently at rate of 50 to 60% per year; 

(e) Interest on cost and depreciation, 16 to 40%. 

To these must be added, to obtain the total cost of haul, 
(J) Wages of the gang employed in shifting track. 



144 EAILROAD CONSTRUCTION. § 110. 

The above calculation for the number of train loads depends 
on the assumption that 9 minutes is total time lost by a 
locomotive for each round trip. If the haul is very short it 
may readily happen that a steam-shovel cannot fill one train 
of cars before the locomotive has returned with a load of empties 
and is ready to haul a loaded train away. The estimation of 
the number of train loads is chiefly useful in planning 
the work so as to have every tool working at its high- 
est efficiency. Usually the capacity of the steam-shovel 
or the ability to promptly "spot'' the cars under the 
shovel is the real limiting agent which determines the daily 
output. 

no. Choice of method of haul dependent on distance. In 
light side-hill work in which material need not be moved more 
than 12 or 15 feet, i.e., moved laterally across the roadbed, 
the earth may be moved most cheaply by mere shovelling. 
Beyond 12 feet scrapers are more economical. At about 100 
feet drag-scrapers and wheelbarrows • are equally economical. 
Between 100 and 200 feet wheelbarrows are generally cheaper 
than either carts or drag-scrapers, but wheeled scrapers are 
always cheaper than wheelbarrows. Beyond 500 feet two- 
wheeled carts become the most economical up to about 1700 
feet; then four-wheeled wagons become more economical up to 
3500 feet. Beyond this <;ars on rails, drawn by horses or by 
locomotives, become cheaper. The economy of cars on rails 
becomes evident for distances as small as 300 feet provided the 
volume of the excavation will justify the outlay. Locomotives 
will always be cheaper than horses and mules, providing the 
work to be done is of sufficient magnitude to justify the pur- 
chase of the necessary plant and risk the loss in selling the plant 
ultimately as second-hand equipment, or keeping the plant on 
hand and idle for an indefinite period waiting for other 
work. Horses will not be economical for distances much 
over a mile. For greater distances locomotives are more 
economical, but the question of ''limit of profitable haul'' 
(§ 116) must be closely studied, as the circumstances are 
certainly not common when it is advisable to haul material 
much over a mile. 

III. Item 4. Spreading. The cost of spreading varies with 
the method employed in dumping the load. When the earth is 



§ 112. EARTHWORK. 145 

tipped over the edge of an embankment there is little if any 
necessary work. Trautwine allows about I c. per cubic yard 
for keeping the dumping-places clear and in order. This would • 
represent the wages of one man at $1 per day attending to the 
unloading of 1200 two-wheeled carts each carrying J cubic yard. 
1200 carts in 10 hours would mean an average of two per minute, 
which implies more rapid and efficient work than may be de- 
pended on. The allowance is probably too small. When the 
material is dumped in layers some levelling is required, for 
which Trautwine allows 50 to 100 cubic yards as a fair day 'a 
work, costing from 1 to 2 cents per cubic yard. The cost of 
spreading will not ordinarily exceed this and is frequently 
nothing — all depending on the method of unloading. It should 
be noted that Mr. Morris's examples and computations (Jour. 
Franklin Inst., Sept. 1841) disregard altogether any special 
charge for this item. 

112. Item 5. Keeping Roadways in order. This feature 
is important as a measure of true economy, whatever the system 
of transportation, but it is often neglected. A petty saving in 
such matters will cost many times as much in increased labor 
in hauling and loss of time. With some methods of haul the 
cost is best combined with that of other items. 

(a) Wheelbarrows. Wheelbarrows should generally be run 
on planks laid on the ground. The adjusting and shifting of 
these planks is done by the wheelers, and the time for it is 
allowed for in the '^f minute for short rests, adjusting the 
wheeling plank, etc." The actual cost of the planks must be 
added, but it would evidently be a very small addition per cubic 
yard in a large contract. When the wheelbarrows are run on 
planks placed on '^horses'' or on trestles the cost is very appre- 
ciable; but the method is frequently used with great economy. 
The variations in the requirements render any general estimate 
of such cost impracticable. 

(b) Carts and wagons. The cost of keeping roadways in 
order for carts and wagons is sometimes estimated merely as so 
much per cubic yard, but it is evidently a function of the lead. 
The work consists in draining off puddles, filling up ruts, pick- 
ing up loose stones that may have fallen off the loads, and in 
general doing everything that will reduce the traction as much 
as possible. Temporary inclines, built to avoid excessive grade 



146 RAILROAD CONSTRUCTION. § 112a. 

at some one pointy are often measures of true economy. Traut- 
wine suggests -^\ c. per cubic yard per 100 feet of lead for earth- 
.work and ^^ c. for rockwork, as an estimate for this item when 
carts are used. 

(c) Cars. When cars are used a shifting-gang, consisting 
of a foreman and several men (say five), are constantly em- 
ployed in shifting the track so that the material may be loaded 
and unloaded where it is desired. The average cost of this 
item may be estimated by dividing the total daily cost of this 
gang by the number of cubic yards handled in one day. 

1 1 2a. Item 6. Trimming cuts to their proper cross- 
section. This process, often called "sand-papering," must 
be treated as an expense, since the payment received for the 
very few cubic yards of earth excavated is wholly inadequate 
to pay for the work involved. Gillette quotes bids of 2 cents 
per square yard of surface trimmed, and from this argues that, 
for average excavations, it adds to the cost four cents per cubic 
yard of the total excavation. The shallower the cut the greater 
is the proportionate cost. Of course the actual cost to the 
contractor will depend largely on the accuracy of outline de- 
manded by the engineer or inspector. 

113. Item 7. Repairs, wear, deprecution, and interest 
on cost of plant. The amount of this item evidently depends 
upon the character of the soil — the harder the soil the worse the 
wear and depreciation. The interest on cost depends on the 
current borrowing value of money. The estimate for this item 
has already been included in the allowances for horses, carts, 
ploughs, harness, wheelbarrows, steam-shovels, etc. Trautwine 
estimates J c. per cubic yard for picks and shovels. Deprecia- 
tion is generally a large percentage of the cost of earth-working 
ijtools, the life of all being limited to a few years, and of many 
tools to a few months. 

114. Item 8. Superintendence and incidentals. The 
incidentals include the cost of water-boys, timekeepers, watch* 
men, blacksmiths, fences, and other precautions to protect the 
public from possible injury, cost of casualty insurance for 
workmen, etc. Although the cost of some of these sub-items 
may be definitely estimated, others are so uncertain that it is 
only possible to make a lump estimate and add say 5 to 7% 
of the sum of the previous items for this item. 



§ 115. EARTHWORK. 147 

115. Contractor's profit and contingencies. The word ''con- 
tingencies'^ here refers to the abnormal expenses caused by- 
freshets, continued wet weather, and ''hard luck,'' as dis- 
tinguished from mere incidentals which are really normal 
expenses. They are the expenses which literally cannot be 
foreseen, and on which the contractor must "take chances." 
They are therefore included with the expected profit. The 
allow^ance for these two elements combined is variously esti* 
mated up to 25% of the previously estimated cost of the work,' 
according to the sharpness of the competition, the contractor's 
confidence in the accuracy of his estimates, and the possible un- 
certainty as to true cost owing to unfavorable circumstances. 
The contractor's real profit may vary considerably from this. 
He often pays clerks, boards and lodges the laborers in shan- 
ties built for the purpose, or keeps a supply-store, and has 
various other items both of profit and expense. His profit 
is largely dependent on skill in so handling the men that all 
can work effectively without interference or delays in wait- 
ing for others. An unusual season of bad weather will often 
affect the cost very seriously. It is a common occurrence 
to find that two contractors may be working on the same kind 
of material and under precisely similar conditions and at the 
same price, and yet one rmay be making money and the 
other losing it — all on account of difference of manage- 
ment. 

116. Limit of profitable haul. As intimated in §§ 103 and 
110, there is with every method of haul a limit of distance be- 
yond which the expense for excessive hauling will exceed the 
loss resulting from borrowing and wasting. This distance is 
somewhat dependent on local conditions, thus requiring an inde-' 
pendent solution for each particular case, but the general prin- 
ciples involved will be about as follows : Assume that it has been 
determined, as in Fig. 62, that the cut and fill will exactly bal- 
ance between two points, as between e and x, assuming that, as 
indicated in § 101 (9), a trestle has been introduced between s 
and t, thus altering the mass curve to Estxn . . . Since there 
is a balance between A' and C, the material for the fill between 
C and e' must be obtained either by "borrowing" in the im- 
mediate neighborhood or by transportation from the excavation 



148 EAILROAD CONSTRUCTION. § 116. 

between 2' and n\ If cut and fill have been approximately 
balanced in the selection of grade line, as is ordinarily don©^ 
borrowing material for the fill C'e^ implies a wastage of material 
at the cut z'rJ . To compare the two methods, we may place 
against the plan of borrowing and wasting, (a) cost, if any, of 
extra right of way that may be needed from which to obtain 
earth for the fill C'e'\ (h) cost of loosening, loading, hauling 
a distance equal to that between the centers of gravity of the 
borrow-pit and of the fill, and the other expenses incidental to 
borrowing M cubic yards for the fill Ce'; (c) cost of loosening, 
loading, hauling a distance equal to that between the centers 
of gravity of the cut z'n^ and of the spoil-bank, and the other 
expenses incidental to wasting M cubic yards at the cut z'n^: 
(d) cost, if any, of land needed for the spoil-bank. The cost of 
the other plan will be the cost of loosening, loading, hauling (the 
hauling being represented by the trapezoidal figure Cexn), and 
the other expenses incidental to making the fill Ce' with the ' 
material from the cut z^n^, the amount of material being M cubic 
yards, which is represented in the figure by the vertical ordi- 
nate from e to the line Cn. The difference between these costs 
will be the cost, if any, of land for borrow-pit and spoil-bank 
plus the cost of loosening, loading, etc. (except hauling and 
roadways) of M cubic yards, minus the difference in cost of the 
excessive haul from Ce to xn and the comparatively short hauls 
from borrow-pit and to spoil-bank. 

As an illustration, taking some of the estimates previously 
given for operating with average material, the cost of all items, 
except hauling and roadways, would be about as follows: 
loosening, with plough, 1.2 c, loading 5.0 c, spreading 1.5 c, 
wear, depreciation, etc., .25 c, superintendence, etc., 1.5 c; 
total 8.95 c. Suppose that the haul for both borrowing and 
wasting averages 100 feet or 1 station. Then the cost of haul 
per yard, using carts, would be (§ 109, a) [125X3(l+4)]4-600 
= 3.125 c. The cost of roadways would be about 0.1 c. per yard, 
making a total of 3.225 c. per cubic yard. Assume ikf = 10000 
cubic yards and the area (7ea;n = 180000 yards-stations or the 
equivalent of 10000 yards hauled 1800 feet. This haul would 
cost [125X3(18 + 4)]-v-600 = 13.75 c. per cubic yard. The cost 
of roadways will be 18 X .1 or 1.8 c, making a total of 15.55 c. for 
hauling and roadways. The difference of cost of hauling and 
roadways will be 15.55-(2X3.225) =9.10 c. per yard or $910 



§ 117. EAKTHWORK. 149 

for the 10000 yards. Offsetting this is the cost of loosening, etc,, 
10000 yards, at 8.95 c, costing S895. These figures may be 
better compared as follows : 

f Loosening, etc., 10000 yards, @, 8.95 c. $ 895. 



^ j Hauling, '* 10000 *' (^ 15.55 c. 

Long Haul. ^ 

[ 



1555. 

$2450. 





'Loosening, etc. 


, 10000 yards (borrowed) 


@ 8.95 c. S895. 




" *' 


10000 " (wasted), 


@ 8.95 c. 895. 


Borrowing 


Hauling, etc., 


10000 " (borrowed) 


@ 3.225 c. 322.50 


AND 

Wasting. 


. 


10000 " (wasted), 


@ 3.225 c. 322.50 
$2435.00 



These costs are practically balanced, but no allowance has 
been made for right of way. If any considerable amount had 
to be paid for that, it would decide this particular case in favor 
of the long haul. This shows that under these conditions 1800 
feet is about the limit of profitable haul, the land costing nothing 
extra. 

BLASTING. 

117. Explosives. The effect of blasting is due to the ex- 
tremely rapid expansion of ^ gas which is developed by the 
decomposition of a very small amount of solid matter. Blasting 
compounds may be divided into two general classes, (a) slow- 
burning and {h) detonating. Gunpowder is a type of the slow- 
burning compounds. These are generally ignited by heat; the 
ignition proceeds from grain to grain; the heat and pressure 
produced are comparatively low. Nitro-glycerine is a type of 
the detonating compounds. They are exploded by a shock 
which instantaneously explodes the whole mass. The heat and 
pressure developed are far in excess of that produced by the 
explosion of powder. Nitro-glycerine is so easily exploded 
that it is very dangerous to handle. It was discovered that if 
the nitro-glycerine was absorbed by a spongy material like infu- 
sorial earth, it ^as much less liable to explode, while its power 
when actually exploded was practically equal to that of the 
amount of pure nitro-glycerine contained in the dynamite, which 
is the name given to the mixture of nitro-glycerine and infusorial 
earth. Nitro-glycerine is expensive; many other explosive 
chemical compounds which properly belong to the slow-burning 



150 EAILROAD CONSTRUCTION. § 118. 

class are comparatively cheap. It has been conclusively demon- 
strated that a mixture of nitro-glycerine and some of the cheaper 
chemicals has a greater explosive force than the sum of the 
strengths of the component parts when exploded separately. 
Whatever the reason, the fact seems established. The reason is 
possibly that the explosion of the nitro-glycerine is sufficiently 
powerful to produce a detonation of the other chemicals, which 
is impossible to produce by ordinary means, and that this explo- 
sion caused by detonation is more powerful than an ordinary 
explosion. The majority of the explosive compounds and 
'* powders" on the market are of this character — a mixture of 
20 to 60 per cent, of nitro-glycerine with variable proportions of 
one or more of a great variety of explosive chemicals. 

The choice of the explosive depends on the character of the 
rock. A hard brittle rock is most effectively blasted by a 
detonating compound. The rapidity with which the full force 
of the explosive is developed has a shattering effect on a brittle 
substance. On the contrary, some of the softer tougher rocks 
and indurated clays are but little affected by dynamite. The 
result is but little more than an enlargement of the blast-hole. 
Quarrying must generally be done with blasting-powder, as the 
quicker explosives are too shattering. Although the results 
obtained by various experimenters are very variable, it may be 
said that pure nitro-glycerine is eight times as powerful as black 
powder, dynamite (75% nitro-glycerine) six times, and gun- 
cotton four to six times as powerful. For open work where 
time is not particularly valuable, black powder is by far the 
cheapest, but in tunnel-headings, whose progress determines the 
progress of the whole work, dynamite is so much more effective 
and so expedites the work that its use becomes economical. 

1 1 8. Drilling. Although many very complicated forms of 
drill-bars have been devised, the best form (with slight modifi- 
cations to suit circumstances) is as shown in Fig. 64, (a) and (6). 
The width should flare at the bottom (a) about 15 to 30%. For 
hard rock the curve of the edge should be somewhat flatter and 
for soft rock somewhat more curved than shown. Fig. 64, (a). 
Sometimes the angle of the two faces is varied from that given, 
Fig. 64, (h), and occasionally the edge is purposely blunted so 
as to give a crushing rather than a cutting effect. The drills 
will require sharpening for each 6 to 18 inches depth of hole, 
and will require a new edge to be worked every 2 to 4 days. 



§119. 



EARTHWORK. 



151 



For drilling vertical holes the churn-drill is the most econom- 
ical. The drill-bar is of iron, about 6 to 8 feet long, l-J" in 
diameter, weighs about 25 to 30 lbs., and is shod with a piece 
of steel welded on. The bar is lifted a few inches between each 
blow, turned partially around, and allowed to fall, the impact 
doing the work. From 5 to 15 feet of holes, depending on the 
character of the rock, is a fair day's work — 10 hours. In very 
soft rocks even more than this may be done. This method is 





Fig. 64. 

inapplicable for inclined holes or even for vertical holes in con- 
fined places, such as tunnel-headings. For such places the only 
practical hand method is to use hammers. This may be done 
by light drills and hght hammers (one-man work), or by heavier 
drills held by one man and struck by one or two men T\ith heavy 
hammers. The conclusion of an exhaustive investigation as to 
the relative economy of light or heavy hammers is that the light- 
hammer method is more economical for the softer rocks, the 
heavy-hammer method is more economical for the harder rocks, 
but that the light-hammer method is always more expeditious 
and hence to be preferred when time is important. 

The subject of machine rock-drills is too vast to be treated 
here. The method is only practicable when the amount of 
work to be done is large, and especially when time is valuable. 
The machines are generally operated by compressed air for tun- 
nel-work, thus doing the additional service of supplying fresh 
air to the tunnel-headings where it is most needed. The cost 
per foot of hole drilled is quite variable, but is usually some- 
what less than that of hand-drilling — sometimes but a smaU 
fraction of it. 

119. Position and direction of drill-holes. As the cost of 
drilling holes is the largest single item in the total cost of blast- 
ing, it is necessary that skill and judgment should be used in so 



152 



KAlLEOAD CONSTRUCTION. 



§120. 



locating the holes that the blasts will be most effective. The 
greatest effect of a blast will evidently be in the direction of the 
''line of least resistance." In a strictly homogeneous material 
this will be the shortest hne from the center of the explosive to 
the surface. The variations in homogeneity on account of 
laminations and seams require that each case shall be judged 
according to experience. In open-pit blasting it is generally 
easy to obtain two and sometimes three exposed faces to the 

rock, making it a simple matter 
to drill holes so that a blast will 
do effective work. When a solid 
face of rock must be broken into, 
as in a tunnel-heading, the work 
is necessarily ineffectual and ex- 
pensive. A conical or wedge- 
shaped mass will first be blown 
out by simultaneous blasts in 
the holes marked 1, Fig. 65; 
blasts in the holes marked 2 and 
3 will then complete the cross- 
section of the heading. A great saving in cost may often be 
secured by skilfully taking advantage of seams, breaks, and irreg- 
ularities. When the work is economically done there is but little 
noise or throwing of rock, a covering of old timbers and branches 
of trees generally sufficing to confine the smaller pieces which 
would otherwise fly up. 

120. Amount of explosive. The amount of explosive required 
varies as the cube of the line of least resistance. The best 
results are obtained when the line of least resistance is J of the 
depth of the hole; also when the powder fills about ^ of the hole. 
For average rock the amount of powder required is as follows : 




DRILL HOLES IN TUNNEL HEADING 
Fig. 65. 



Line of least resistance. 
Weight of powder 



2 ft. 


4 ft. 


6 ft. 


i lb. 


2 lbs. 


6J lbs. 



8 ft. 
16 lbs. 



Strict compliance with all of the above conditions would re- 
quire that the diameter of the hole should vary for every case. 
While this is impracticable, there should evidently be some 
variation in the size of the hole, depending on the work to be 
done. For example, a 1'' hole, drilled 2' 8" deep, with its 
line of least resistance 2'. and loaded with J lb. of powder, would 



§ 121. EARTHWORK. 153 

be filled to a depth of 9^, which is nearly i of the depth. A 
3'' hole, drilled 8' deep, with its line of least resistance 6', and 
loaded with 6| lbs. of powder, would be filled to a depth of over 
28'', which is also nearly i of the depth. One pound of blasting- 
powder will occupy about 28 cubic inches. Quarrying necessi- 
tates the use of numerous and sometimes repeated light charges of 
powder, as a heavy blast or a powerful explosive like dynamite 
is apt to shatter the rock. This requires more powder to the 
cubic yard than blasting for mere excavation, which may usually 
be done by the use of J to ^ lb. of powder per cubic yard of easy 
open blasting. On account of the great resistance offered by 
rock when blasted in headings in tunnels, the powder used per 
cubic yard will run up to 2, 4, and even 6 lbs. per cubic yard. 
As before stated, nitro-glycerine is about eight times (and 
dynamite about six times) as powerful as the same weight of 
powder. 

121. Tamping. Blasting-powder and the slow-burning ex- 
plosives require thorough tamping. Clay is probably the best, 
but sand and fine powdered rock are also used. Wooden plugs, 
inverted expansive cones, etc., are periodically reinvented by 
enthusiastic inventors, only to be discarded for the simpler 
methods. Owing to the extreme rapidity of the development 
of the force of a nitro-glycerine or dynamite explosion, tamping 
is not so essential with these explosives, although it unquestion- 
ably adds to their effectiveness. Blasting under water has been 
effectively accomplished by merely pouring nitro-glycerine into 
the drilled holes through a tube and then exploding the charge 
without any tamping except that furnished by the superincum- 
bent water. It has been found that air-spaces about a charge 
make a material reduction in the effectiveness of the explosion. 
It is therefore necessary to carefully ram the explosive into a 
solid mass. Of course the liquid nitro-glycerine needs no ram- 
ming, but dynamite should be rammed with a wooden rammer. 
Iron should be carefully avoided in ramming gunpowder. A 
copper bar is generally used. 

122. Exploding the charge. Black powder is generally ex- 
ploded by means of a fuse which is essentially a coTd in which 
there is a thin vein of gunpowder, the cord being protected by 
tar, extra linings of hemp, cotton, or even gutta-percha. The 
fuse is inserted into the middle of the charge, and the tamping 
carefully packed around it so that it will not be injured. To 



154 RAILROAD CONSTRUCTION. § 123. 

produce the detonation required to explode nitro-glycerine and 
dynamite, there must be an initial explosion of some easily 
ignited explosive. This is generally accomphshed by means of 
caps containing fulminating-powder which are exploded by 
electricity. The electricity (in one class of caps) heats a very 
fine platinum wire to redness, thereby igniting the sensitive 
powder, or (in another class) a spark is caused to jump through 
the powder between the ends of two wires suitably separated. 
Dynamite can also be exploded by using a small cartridge of 
gunpowder which is itself exploded by an ordinary fuse. 

123. Cost. Trautwine estimates the cost of blasting (for 
mere excavation) as averaging 45 cents per cubic yard, falling 
as low as 30 cents for easy but brittle rock, and running up to 
60 cents and even $1 when the cutting is shallow, the rock 
especially tough, and the strata unfavorably placed. Soft tough 
rock frequently requires more powder than harder brittle rock. 

124. Classification of excavated material. The classification 
of excavated material is a fruitful source of dispute between 
contractors and railroad companies, owing mainly to the fact 
that the variation between the softest earth and the hardest rock 
is so gradual that it is very difficult to describe distinctions 
between different classifications which are unmistakable and 
indisputable. The classification frequently used is (a) earth, 
(6) loose rock, and (c) solid rock. As blasting is frequently 
used to loosen ''loose rock^' and even ''earth'' (if it is frozen), 
the fact that blasting is employed cannot be used as a criterion, 
especially as this would (if allowed) lead to unnecessary blasting 
for the sake of classifying material as rock. 

Earth. This includes clay, sand, gravel, loam, decomposed 
rock and slate, boulders or loose stones not greater than 1 cubic 
foot (3 cubic feet, P. R. R.), and sometimes even "hard-pan." 
In general it will signify material which can be loosened by a 
plough with two horses, or with which one picker can keep one 
shoveller busy. 

Loose rock. This includes boulders and loose stones of more 
than one cubic foot and less than one cubic yard; stratified rock, 
not more than six inches thick, separated by a stratum of clay; 
also all material (not classified as earth) which may be loosened 
by pick or bar and which ^' can be quarried without blasting, 
although blasting may occasionally be resorted to," 



§ 125, EARTHWORK. 155 

Solid rock includes all rock found in masses of over one cubic 
yard which cannot be removed except by blasting. 

It is generally specified that the engineer of the railroad 
company shall be the judge of the classification of the material, 
but frequently an appeal is taken from his decisions to the 
courts. 

125. Specifications for earthwork. The following specifica- 
tions, issued by the Norfolk and Western R, R., represent the 
average requirements. It should be remembered that very 
strict specifications invariably increase the cost of the work, 
and frequently add to the cost more than is gamed by improved 
quality of work. 

1, The grading will be estimated and paid for by the cubic 
yard, and will include clearing and grubbing, and all open ex- 
cavations, channels, and embankments required for the forma- 
tion of the roadbed, and for turnouts and sidings; cutting all 
ditches or drains about or contiguous to the road; digging the 
foundation-pits of all culverts, bridges, or walls; reconstructing 
turnpikes or common roads in cases where they are destroyed or 
interfered with; changing the course or channel of streams; and 
all other excavations or embankments connected with or incident 
to the construction of said Railroad. 

2. All grading, except where otherwise specified, whether 
for cuts or fills, will be measured in the excavations and will be 
classified under the following heads, viz.: Solid Rock, Loose 
Rock, Hard-pan, and Earth. 

Solid Rock shall include all rock occurring in masses which, 
in the judgment of the said Engineer Maintenance of Way, may 
be best removed by blasting. 

Loose Rock shall include all kinds of shale, soapstone, and 
other rock w^hich, in the judgment of the said Engineer Main- 
tenance of Way, can be removed by pick and bar, and is soft and 
loose enough to be removed without blasting, although blasting 
may be occasionally resorted to ; also, detached stone of less than 
one (1) cubic yard and more than one (1) cubic foot. 

Hard-pan shall consist of tough indurated clay or cemented 
gravel, w^hich requires blasting or other equally expensive 
means for its removal, or which cannot be ploughed with less 
than four horses and a railroad plough, or which requires two 
pickers to a shoveller, the said Engineer Maintenance of Way 
to be the judge of these conditions. 



156 RAILROAD CONSTRUCTION. § 125. 

Earth shall include all material of an earthy nature, of what- 
ever name or, character, not unquestionably loose rock or hard- 
pan as above defined. 

Powder. The use of powder in cuts will not be considered 
as a reason for any other classification than earth, unless the 
material in the cut is clearly other than earth under the above 
specifications. 

3. Earth, gravel, and other materials taken from the exca- 
vations, except when otherwise directed by the said Engineer 
Maintenance of Way or his assistant, shall be deposited in the 
adjacent embankment; the cost of removing and depositing 
which, when the distance necessary to be hauled is not more 
than sixteen hundred (1600) feet, shall be included in the price 
paid for the excavation. 

4. Extra Haul will be estimated and paid for as follows: 
whenever material from excavations is necessarily hauled a 
greater distance than sixteen hundred (1600) feet, there shall be 
paid in addition to the price of excavation the price of extra 
haul per 100 feet, or part thereof, after the first 1600 feet; the 
necessary haul to be determined in each case by the said Engi- 
neer Maintenance of Way or his assistant, from the profile and 
cross-sections, and the estimates to be in accordance therewith. 

5. All embankments shall be made in layers of such thick- 
ness and carried on in such manner as the said Engineer Mainte- 
nance of Way or his assistant may prescribe, the stone and heavy 
materials being placed in slopes and top. And in completing 
the* fills to the proper grade such additional heights and fulness 
of slope shall be given them, to provide for their settlement, as 
the said Engineer Maintenance of Way, or his assistant, may 
direct. Embankments about masonry shall be built at such 
times and in such manner and of such materials as the said Engi- 
neer Maintenance of Way or his assistant may direct. 

6. In procuring materials for embankments from without 
the line of the road, and in wasting materials from cuttings, the 
place and manner of doing it shall in each case be indicated by 
the Engineer Maintenance of Way or his assistant; and care 
must be taken to injure or disfigure the land as little as possible. 
Borrow-pits and spoil-banks must be left by the Contractor in 
regular and sightly shape. 

7. The lands of the said Railroad Company shall be cleared 
to the extent required by the said Engineer Maintenance of 



§ 125. EARTHWORK. 157 

Way, or his assistant, of all trees, brushes, logs, and other perish- 
able materials, which shall be destroyed by burning or deposited 
in heaps as the said Engineer Maintenance of Way, or his assist- 
ant, may direct. Large trees must be cut not more than two 
and one-half (2 J) feet from the ground, and under embank- 
ments less than four (4) feet high they shall be cut close to the 
ground. All small trees and bushes shall be cut close to the 
ground. 

8. Clearing shall be estimated and paid for by the acre or 
fraction of an acre. 

9. All stumps, roots, logs, and other obstructions shall be 
grubbed out, and removed from all places where embankments 
occur less than two (2) feet in height; also, from all places where 
excavations occur and from such other places as the said Engi- 
neer Maintenance of Way or his assistant may direct. 

10. Grubbing shall be estimated and paid for by the acre or 
fraction of an acre, 

11. Contractors, when directed by the said Engineer Main- 
tenance of Way or his assistant in charge of the work, will deposit 
on the side of the road, or at such convenient points as may be 
designated, any stone, rock, or other materials that they may 
excavate; and all materials excavated and deposited as above, 
together with all timber removed from the line of the road, will 
be considered the property of the Railroad Company, and the 
Contractors upon the respective sections will be responsible for 
its safe-keeping until removed by said Railroad Company, or 
until their work is finished. 

12. Contractors will be accountable for the maintenance of 
safe and convenient places wherever public or private roads are 
in any way interfered with by them during the progress of the 
work. They will also be responsible for fences thrown down, 
and for gates and bars left open, and for all damages occasioned 
thereby. 

13. Temporary bridges and trestles, erected to facilitate the 
progress of the work, in case of delays at masonry structures 
from any cause, or for other reasons, will be at the expense of 
the Contractor. 

14. The line of road or the gradients may be changed in any 
manner, and at any time, if the said Engineer Maintenance of 
Way or his assistant shall consider such a change necessary or 
expedient; but no claim for an increase in prices of excavation 



158 RAILROAD CONSTRUCTION. § 125. 

or embankment on the part of the Contractor will be allowed 
or considered unless made in writing before the work on that 
part of the section where the alteration has been made shall have 
been commenced. The said Engineer Maintenance of Way or 
his assistant may also, on the conditions last recited, increase or 
diminish the length of any section for the purpose of more nearly 
equalizing or balancing the excavations and embankments, or 
for any other reason. 

15. The roadbed will be graded as directed by the said En- 
gineer Maintenance of Way or his assistant, and in conformity 
with such breadths, depths, and slopes of cutting and filling as 
he may prescribe from time to time, and no part of the work 
will be finally accepted until it is properly completed and dressed 
off at the required grade. 



CHAPTER IV. 
TRESTLES. 

126. Extent of use. Trestles constitute from 1 to 3% of the 
length of the average railroad. It was estimated in 1889 that 
there was then about 2400 miles of single-track railway trestle 
in the United States, divided among 150,000 structures and esti- 
mated to cost about $75,000,000. The annual charge for main- 
tenance, estimated at J of the cost, therefore amounted to about 
$9,500,000 and necessitated the annual use of perhaps 300,000,000 
ft. B. M. of timber. The corresponding figures at the present 
time must be somewhat in excess of this. The magnitude of 
this use, which is causing the rapid disappearance of forests, has 
resulted in endeavors to limit the use of timber for this purpose. 
Trestles may be considered as justifiable under the following 
conditions : 

a. Permanent trestles. _ 

1. Those of extreme height — then called viaducts and fre- 
quently constructed of iron or steel, as the Kinzua viaduct, 302 
ft. high. 

2. Those across waterways — e.g., that across Lake Pontchar- 
train, near New Orleans, 22 miles long. 

3. Those across swamps of soft deep mud, or across a river- 
bottom, liable to occasional overflow. 

h. Temporary trestles. 

1. To open the road for traffic as quickly as possible — often 
a reason of great financial importance. 

2. To quickly replace a more elaborate structure, destroyed 
by accident, on a road already in operation, so that the inter- 
ruption to traffic shall be a minimum. 

3. To form an earth embankment with earth brought from 
a distant point by the train-load, when such a measure would 
cost less than to borrow earth in the immediate neighborhood. 

4. To bridge an opening temporarily and thus allow time to 
learn the regimen of a stream in order to better proportion the 

159 



160 RAILROAD CONSTRUCTION. § 127. 

size of the waterway and also to facilitate bringing suitable stone 
for masonry from a distance. In a new country there is always 
the double danger of either building a culvert too small, requir- 
ing expensive reconstruction, perhaps after a disastrous washout, 
or else wasting money by constructing the culvert unnecessarily 
large. Much masonry has been built of a very poor quality of 
stone because it could be conveniently obtained and because 
good stone was unobtainable except at a prohibitive cost for 
transportation. Opening the road for traffic by the use of 
temporary trestles obviates both of these difficulties. 

127. Trestles vs. embankments. Low embankments are very- 
much cheaper than low trestles both in first cost and mainte- 
nance. Very high embankments are very expensive to con- 
struct, but cost comparatively little to maintain. A trestle of 
equal height may cost much less to construct, but will be expen- 
sive to maintain — perhaps J of its cost per year. To determine 
the height beyond which it will be cheaper to maintain a trestle 
rather than build an embankment, it will be necessary to allow 
for the cost of maintenance. The height will also depend on 
the relative cost of timber, labor, and earthwork. At the pres- 
ent average values, it will be found that for less heights than 
25 feet the first cost of an embankment will generally be less 
than that of a trestle; this implies that a permanent trestle 
should never be constructed with a height less than 25 feet except 
for the reasons given in § 126. The height at which a permanent 
trestle is certainly cheaper than earthwork is more uncertain. 
A high grade line joining two hills will invariably imply at least 
a culvert if an embankment is used. If the culvert is built of 
masonry, the cost of the embankment will be so increased that 
the height at which a trestle becomes economical will be mate- 
rially reduced. The cost of an embankment increases much 
more rapidly than the height — with very high embankments 
more nearly as the square of the height — while the cost of 
trestles does not increase as rapidly as the height. Although 
local circumstances may modify the application of any set rules, 
it is probably seldom that it will be cheaper to build an embank- 
ment 40 or 50 feet high than to permanently maintain a wooden 
trestle of that height. A steel viaduct would probably be the 
best solution of such a case. These are frequently used for 
permanent structures, especially when very high. The cost of 
maintenance is much less than that of wood, which makes the 



§ 128. TRESTLES. 161 

use of iron or steel preferable for permanent trestles unless wood 
is abnormally cheap. Neither the cost nor the construction 
of iron or steel trestles will be considered in this chapter. 

128. Two principal types. There are two principal types of 
wooden trestles — pile trestles and framed trestles. The great 
objection to pile trestles is the rapid rotting of the portion of the 
pile which is underground, and the difficulty of renewal. The 
maximum height of pile trestles is about 30 feet, and even this 
height is seldom reached. Framed trestles have been con- 
structed to a height of considerably over 100 feet They are 
frequently built in such a manner that any injured piece may be 
readily taken out and renewed without interfering with traffic. 
Trestles consist of two parts — the supports called ^' bents/ ^ and 
the stringers and floor system. As the stringers and floor system 
are the same for both pile and framed trestles, the " bents " are 
all that need be considered separately. 



PILE TRESTLES. 

129, Pile bents. A pile bent consists generally of four piles 
driven into the ground deep enough to afford not only sufficient 
vertical resistance but also lateral resistance. On top of these 
piles is placed a horizontal ^'^ap." The caps are fastened to 
the tops of the piles by methods illustrated in Fig. 66. The 
method of fastening shown in each case should not be considered 
as applicable only to the particular type of pile bent used to illus- 
trate it. Fig. 66 (a and d) illustrates a mortise- joint with a hard- 
wood pin about IJ'' in diameter. The hole for the pin should 
be bored separately through the cap and the mortise, and the 
hole through the cap should be at a slightly higher level than 
that through the mortise, so that the cap will be drawn down 
tight when the pin is driven. Occasionally an iron dowel (an 
iron pin about IJ'' in diameter and about 6'' long) is inserted 
partly in the cap and partly in the pile. The use of drift-bolts, 
shown in Fig. 66 (6), is cheaper in first cost, but renders repairs 
and renewals very troublesome and expensive. ^' Split caps,'' 
shown in Fig. 66 (c), are formed by bolting two half-size strips 
on each side of a tenon on top of the pile. Repairs are very 
easily and cheaply made without interference with the traffic 
and without injuring other pieces of the bent. The smaller 
pieces are more easily obtainable in a sound condition; the 



162 



RAILROAD CONSTRUCTION. 



129. 



decay of one does not affect the other, and the first cost is but 
little if any greater than the method of using a single piece. For 
further discussion, see § 136. 

For very light traffic and for a height of about 5 feet three 
vertical piles will suffice, as shown in Fig. 66 (a)o Up to a height 




Fig. 66. 

of 8 or 10 feet four piles may be used without sway-bracing, as 
in Fig 66 (h), if the piles have a good bearing. For heights 
greater than 10 feet sway-bracing is generally necessary. The 
outside piles are frequently driven with a batter varying from 
1 : 12 to 1 : 4. 

Piles are made, if possible, from timber obtained in the 
vicinity of the work. Durability is the great requisite rather 
than strength, for almost any timber is strong enough (except 
as noted below) and will be suitable if it will resist rapid decay. 
The following list is quoted as being in the order of preference 
on account of durability: 



1. Red cedar 

2. Red cypress 

3. Pitch-pine 

4. Yellow pine 



5. White pine 

6. Redwood 

7. Elm 

8. Spruce 



9. White oak 

10. Post-oak 

11. Red oak 



12. Black oak 

13. Hemlock 

14. Tamarac 



Red-cedar piles are said to have an average life of 27 years 
with a possible maximum of 50 years, but the timber is rather 



§ 130. TKESTLES. 163 

weak, and if exposed in a river to floT\ang ice or driftwood is 
apt to be injured. Under these circumstances oak is prefer- 
able, although its life may be only 13 to 18 years. 

130. Methods of driving piles. The following are the prin- 
cipal methods of driving piles : 

a. A hammer weighing 2000 to 3000 lbs. or more, sliding 
in guides, is drawn up by horse-power or a portable engine, and 
allowed to fall freely. 

h. The same as above except that the hammer does not fall 
freely, but drags the rope and revohang drum as it falls and is 
thus quite materially retarded. The mechanism is a Httle more 
simple, but is less effective, and is sometimes made deliberately 
deceptive by a contractor by retarding the blow, in order to 
apparently indicate the requisite resistance on the part of 
the pile. 

The above methods have the advantage that the mechanism 
is cheap and can be transported into a new country with com- 
parative ease, but the work done is somewhat ineffective and 
costly compared with some of the more elaborate methods 
given below. 

c. Gunpowder pile-drivers^ which automatically explode a 
cartridge every time the hammer falls. The explosion not only 
forces the pile down, but throws up the hammer for the next 
blow. For a given height of fall the effect is therefore doubled. 
It has been shown by experience, however, that when it is at- 
tempted to use such a pile-driver rapidly the mechanism be- 
comes so heated that the cartridges explode prematurely, and the 
method has therefore been abandoned. 

d. Steam pile-drivers, in which the hammer is operated 
directly by steam. The hammer falls freely a height of about 
40 inches and is raised again by steam. The effectiveness is 
largely due to the rapidity of the blows, which does not allow 
time between the blows for the ground to settle around the pile 
and increase the resistance, which does happen when the blows 
are infrequent. '^The hammer-cylinder weighs 5500 lbs., and 
with 60 to 75 lbs. of steam gives 75 to 80 blows per minute. 
With 41 blows a large unpointed pile was driven 35 feet into a 
hard clay bottom in half a minute.'' Such a driver would cost 
about $800. 

The above four methods are those usual for dry earth. In 
very soft wet or sandy soils, where an unlimited supply of water 



164 EAILROAD CONSTRUCTION. § 13x. 

is available, the water-jet is sometimes employed. A pipe is 
fastened along the side of the pile and extends to the pile-point. 
If water is forced through the pipe^ it loosens the sand around 
the point and, rising along the sides, decreases the side resist- 
ance so that the pile sinks by its own weight, aided perhaps by 
extra weights loaded on. This loading may be accomplished by 
connecting the top of the pile and the pile-driver by a block 
and tackle so that a portion of the weight of the pile-driver is 
continually thrown on the pile. 

Excessive driving frequently fractures the pile below the 
surface and thereby greatly weakens its bearing power. To 
prevent excessive '^brooming" of the top of the 
pile, owing to the action of the hammer, the top 
should be. protected by an iron ring fitted to the 
top of the pile. The '^brooming" not only ren- 
ders the driving ineffective and hence uneconomi- 
cal, but vitiates the value of any test of the bearing 
power of the pile by noting the sinking due to a 
given weight falling a given distance. If the pile 
is so soft that brooming is unavoidable, the top 
Fig. 67. should be adzed off frequently, and especially 
should it be done just before the final blows which are to test its 
bearing-power. 

In a new country judgment and experience will be required 
to decide intelligently whether to employ a simple drop-hammer 
machine, operated by horse-power and easily transported but 
uneconomical in operation, or a more complicated machine 
working cheaply and effectively after being* transported at 
greater expense. 

131. Pile- driving formulae. If R=the resistance of a pile, 
and s the set of the pile during the last blow, w the weight of 
the pile-hammer, and h the fall during the last blow, then we 

may state the approximate relation that R6=whj or R = ~, 

This is the basic principle of all rational formulae, but the maxi- 
mum weight which a pile will sustain after it has been driven 
some time is by no means equal to the resistance of the pile 
during the last blow. There are also many other modifying 
elements which have been variously allowed for in the many 
proposed formulae. The formulae range from the extreme of 
empirical simplicity to very complicated attempts to allow 




§ 131. TRESTLES. 165 

properly for all modifying causes. As the simplest rule, speci- 
fications sometimes require that the piles shall be driven until 
the pile will not sink more than 5 inches under five consecutive 
blows of a 2000-lb. hammer falling 25 feet. The ^'Engineering 

News formula" * gives the safe load as --. in which w = 

weight of hammer, /t=fall in feet, s=set of pile in inches under 
the last blow. This formula is derived from the above basic 
formula by calling the safe load \ of the final resistance, and 
by adding (arbitrarily) 1 to the final set (s) as a compensation 
for the extra resistance caused by the settling of earth around 
the pile between each blow. This formula is used only for 
ordinary hammer-driving. When the piles are driven by a 

steam pile-driver the formula becomes safe load = 7--. For 

^ s + 0.1 

the "gunpowder pile-driver," since the explosion of the cartridge 

drives the pile in with the same force with which it throws the 

hammer upward, the effect is twice that of the fall of the hammer, 

and the formula becomes safe load = --^. In these last two 

s + 0.1 

formulae the constant in the denominator is changed from s + 1 

to s + 0.1. The constant (1.0 or 0.1) is supposed to allow, as 

before stated, for the effect of-^he extra resistance caused by the 

earth settling around the pile between each blow. The more 

rapid the blows the less the opportunity to settle and the less 

the proper value of the constant. 

The above formulae have been given on account of their 
simplicity and their practical agreement with experience. Many 
other formulae have been proposed, the majority of which are 
more complicated and attempt to take into account the weight of 
the pile, resistance of the guides, etc. While these elements, 
as well as many others, have their influence, their effect is so 
overshadowed by the indeterminable effect of other elements — 
as, for example, the effect of the settlement of earth aroimd the 
pile between blows — that it is useless to attempt to employ any- 
thing but a purely empirical formula. 

Examples, 1. A pile was driven with an ordinary hammer 
weighing 2500 pounds until the sinking under five consecutive 
blows was 15i inches. The fall of the hammer during the last 



* Engineering News, Nov. 17, 1892, 



166 



RAILROAD CONSTRUCTION. 



§ 132. 



blows was 24 feet. What was the safe bearing power of the 
pile? 

2wh 2X2500X24 120000 ^^^^^ 

."+1 = (JX15.5) + 1=-T1- =^^^^^ P^^^^'- 

2. Piles are being driven into a firm soil with a steam pile- 
driver until they show a safe bearing power of 20 tons. The 
hammer weighs 5500 pounds and its fall is 40 inches. What 
should be the sinking under the final blow? 



40000= 



2wh 2X5500X3.33 
s + 0.1 ' 

-0.1 =.81 inch. 



s + 0.1 

36667 
' 40000 



132. Pile-points and pile-shoes. Piles are generally sharpened 
to a blunt point. If the pile is liable to strike boulders, sunken 

logs, or other obstructions which are 
liable to turn the point, it h necessary 
to protect the point by some form of 
shoe. Several forms in cast iron have 
been used, also a wrought-iron shoe, 
having four ^'straps" radiating from 
the apex, the straps being nailed on to 
the pile, as shown in Fig. 68 (6). The 
cast-iron form shown in Fig. 68 (a) 
has a base cast around a drift-bolt. 
The recess on the top of the base re- 
FiG. 68. ceives the bottom of the pile and pre- 

vents a tendency to split the bottom of the pile or to force the 
shoe off laterally. 

133. Details of design. No theoretical calculations of the 
strength of pile bents need be attempted on account of the ex- 
treme complication of the theoretical strains, the uncertainty as 
to the real strength of the timber used, the variability of that 
strength with time, and the insignificance of the economy that . 
would be possible even if exact sizes could be computed. The 
piles are generally required to be not less than 10'' or 12" in 
diameter at the large end. The P. R. R. requires that they shall 




§ 134. TRESTLES * 167 

be " not less than 14 and 7 inches in diameter at butt and small 
end respectively, exclusive of bark, which must be removed." 
The removal of the bark is generally required in good work. 
Soft durable woods, such as are mentioned in § 129, are best 
for the piles, but the caps are generally made of oak or yellow 
pine. The caps are generally 14 feet long (for single track) 
with a cross-section 12''Xl2'' or 12"Xl4". ''Split caps" 
would consist of two pieces 6"Xl2''. The sway-braces, never 
used for less heights than 6', are made of 3" X 12'' timber, and 
are spiked on with f '' spikes 8'' long. The floor system will be 
the same as that described later for framed trestles. 

134. Cost of pile trestles. The cost, per Hnear foot, of piling 
depends on the method of driving, the scarcity of suitable tim- 
ber, the price of labor, the length of the piles, and the amount 
of shifting of the pile-driver required. The cost of soft-wood 
piles varies from 8 to 15 c. per lineal foot, and the cost of oak 
piles varies from 10 to 30 c. per foot according to the length, 
the longer piles costing more per foot. The cost of driving T\dll 
average about $2.50 per pile, or 7.5 to 10 c. per lineal foot. 
Since the cost of shifting the pile-driver is quite an item in the 
total cost, the cost of driving a long pile would be less per foot 
than for a short pile, but on the other hand the cost of the pile 
is greater per foot, which tenda to make the total cost per foot 
constant. Specifications generally say that the piling will be 
paid for per lineal foot of piling left in the work. The wastage 
of the tops of piles sawed off is always something, and is fre- 
quently very large. Sometimes a small amount per foot of 
piling sawed off is allowed the contractor as compensation for 
his loss. This reduces the contractor's risk and possibly reduces 
his bid by an equal or greater amount than the extra amount 
actually paid him. 

FRAMED TRESTLES. 

I35« Typical design. A typical design for a framed trestle 
bent is given in Fig. 69. This represents, with slight variations 
of detail, the plan according to which a large part of the framed 
trestle bents of the country have been built — i.e., of those less 
than 20 or 30 feet in height, not requiring multiple story con- 
struction. 

136. Joints, (a) The mortise-and-tenon joint is illustrated in 



168 RAILROAD CONSTRUCTION. § 136. 

Fig. 69 and also in Fig. 66 (a). The tenon should be about 



1 ©>§; 


To"! 


CAP 12"x 12"x 14 


rs-. r:il'0 1! 




/N 


^^-3-- 




rl'fi" 




"/T?i 












/ / 




@ 




''(3) 


'^\\ 




/ 5? / 




\ 






\ \ 




/ / 






>». /N. y 




\ \ 














\ \ 




/ o / 






/^\ ''''^N. 




\ \ 




/ a / 










\ \ 




1^1 










\ \ 




/ o / 




y* 




\, 


\ I 




/ t / 
1^1 




© 




h® 


\ \ \ 








y 




N 


.^N, \ \ 




S y 












i 


1 'I .^^ 




h 






NT'^PiX 


\ 


A \X 




2 






\i 


~^v 


/@y^-^ 


L?-i 


SILL 12"x 12" 


L°-l ^ 


^@\ 



3" thick, 8'^ 



Fig. 70. 



Fig. 69. 

wide, and 51" long. The mortise should be cut 
a little deeper than the tenon. '' Drip-holes'* 
from the mortise to the outside will assist in 
draining off water that may accumulate in the 
joint and thus prevent the rapid decay that 
would otherwise ensue. These joints are very 
troublesome if a single post decays and requires 
renewal. It is generally required that the mor- 
tise and tenon should be thoroughly daubed 
with paint before putting them together. This will tend to 
make the joint water-tight and prevent decay from the accu- 
mulation and retention of water in the joint. 

(b) The plaster joint. This joint is made by bolting and 
spiking a 3''Xl2" plank on 
both sides of the joint. The 
cap and sill should be 
notched to receive the posts. 
Repairs are greatly facili- 
tated by the use of these 
joints. This method has been 
used by the Delaware and 
Hudson Canal Co. [R. R.]. 

(c) Iron plates. An iron plate of the form shown in Fig. 72 




Fig. 71. 



§137. 



TKESTLES. 



169 




,'"'" 


o o 


b 

c 


o 
o 




o 


a 


o o 


(&) " 

a 



Fig. 72. 



(b) is bent and used as shown in Fig. 72 (a). Bolts passing 
through the bolt - holes ^'T~""V'"'i 

shown secure the plates 
to the timbers and make 
a strong joint which may 
be readily loosened for re- 
pairs. By slight modifi- 
cations in the design the | ~" 
method may be used for |^^ 
inclined posts and compli- *" 
cated joints. 

(d) Split caps and sills. 
These are described in 
§ 129. Their advantages apply with even greater force to 
framed trestles. 

(e) Dowels and drift-bolts. These joints facilitate cheap and 
rapid construction, but renewals and repairs are very difficult, it 
being almost impossible to extract a drift-bolt, which has been 
driven its full length, without splitting open the pieces contain- 
ing it. Notwithstanding this objection they are extensively 
used, especially for temporary work which is not expected to 
be used long enough to need repairs. 

— 137. Multiple-story construc- 
tion. Single-story framed trestle 
bents are used for heights up 
to 18 or 20 feet and exception- 
ally up to 30 feet. For greater 
heights some such construction as 
is illustrated in a skeleton design 
in Fig. 73 is used. By using split 
sills between each story and sepa- 
rate vertical and batter posts in 
each story, any piece may readily 
be removed and renewed if neces- 
sary. The height of these stories 
varies, in different designs, from 
15 to 25 and even 30 feet. In 
some designs the structure of each 
story is independent of the stories 
above and below. This greatly 
facilitates both the original construction and subsequent repairs. 




Fig. 73. 



170" 



RAILROAD CONSTRUCTION. 



§138. 



In other designs the verticals and batter-posts are made con- 
tinuous through two consecutive stories. The structure is 
somewhat stiffer, but is much more difficult to repair. 

Since the bents of any trestle are usually of variable height 
and those heights are not always an even multiple of the uniform 
height desired for the stories, it becomes necessary to make the 




Fig. 74. 

upper stories of uniform height and let the odd amount go to the 
lowest story, as shown in Figs. 73 and 74. 

138. Span. The shorter the span the greater the number of 
trestle bents; the longer the span the greater the required strength 
of the stringers supporting the floor. Economy demands the 
adoption of a span that shall make the sum of these require- 




FiQ. 75. 



ments a minimum. The higher the trestle the greater the cost 
of each bent, and the greater the span that would be justifiable. 
Nearly all trestles have bents of variable height, but the advan- 
tage of employing uniform standard sizes is so grea^ that many 



139. 



TRESTLES. 



171 



roads use the same span and sizes of timber not only for the 
panels of any given trestle, but also for all trestles regardless of 
height. The spans generally used vary from 10 to 16 feet. The 
Norfolk and Western R. R. uses a span of 12' 6" for all single- 
story trestles, and a span of 25' for all multiple-story trestles. 
The stringers are the same in both cases, but when the span is 
25 feet, knee-braces are run from the sill of the first story below 
to near the middle of each set of stringers. These knee-braces' 
are connected at the top by a "straining-beam" on which the 
stringers rest, thus supporting the stringer in the center and vir- 
tually reducing the span about one-half. 

139. Foundations, (a) Piles. Piles are frequently used as a 
foimdation, as in Fig. 76, particularly in soft ground, and also 
for temporary structures. These 
foundations are cheap, quickly 
constructed, and are particularly 
valuable when it is financially 
necessary to open the road for 
traffic as soon as possible and 
with the least expenditure of 
money; but there is the disad- 
vantage of inevitable decay 
within a few years unless the piles are chemically treated, as will 
be discussed later. Chemical treatment, however, increases the 
cost so that such a foundation would often cost more than a 
foundation of stone. A pile should be driven under each post 
as shown in Fig. 76. 

(b) Mud-sills. Fig. 




Fig. 76. 



, n w v\ 



.:;-::-.:::51:::::i::^ 


1 SILL 1 


r..:-.::-:....::::!;:: 



Fig. 77. 

(c) Stone foundations, 
the most expensive. 



77 illustrates the use of mud-sills as 
built by the Louisville and 
Nashville R. R. Eight blocks 
12"X12"X6' are used under 
each bent. When the ground 
is very soft, two additional 
timbers (12" X 12" X length of 
bent-sill), as shown by the 
dotted lines, are placed under- 
neath. The number required 
evidently depends on the na- 
ture of the ground. 
Stone foundations are the best and 

For very high trestles the Norfolk and 



172 



EAILEOAD CONSTRUCTION. 



§140. 



Western R. R. employs foundations as shown in Fig. 78, the 
walls being 4 feet thick. When the height of the trestle is 72 
feet or less (the plans requiring for 72' in height a foundation- 
wall 39' 6" long) the foundation is made continuous. The sill 




SILL OF TRESTLg 





* 13 >• ^ ■< 8 *■ "< 7l3 ^ 

Fig. 78. 

of the trestle should rest on several short lengths of 3" X 12" 
plankj laid transverse to the sill on top of the wall. 

140. Longitudinal bracing. This is required to give the 
structure longitudinal stiffness and also to reduce the columnar 
length of the posts. This bracing generally consists of hori- 
zontal "waling-strips" and diagonal braces. Sometimes the 
braces are placed wholly on the outside posts unless the trestle 
is very high. For single-story trestles the P. R. R. employs 
the '4aced" system, i.e., a line of posts joining the cap of one 
bent with the sill of the next, and the sill of that bent with the 
cap of the next. Some plans employ braces forming an X in 
alternate panels. Connecting these braces in the center more 
than doubles their columnar strength. Diagonal braces, when 
bolted to posts, should be fastened to them as near the ends of 
the posts as possible. The sizes employed vary largely, depend- 
ing on the clear length and on whether they are expected to act 
by tension or compression. 3"Xl2" planks are often used 
when the design would require tensile strength only, and 8" X 8" 
posts are often used when compression may be expected. 

141. Lateral bracing. Several of the more recent designs of 
trestles employ diagonal lateral bracing between the caps of 
adjacent bents. It adds greatly to the stiffness of the trestle 
and better maintains its alignment. 6"X6" posts, forming 
an X and connected at the center, will answer the purpose. 

142. Abutments. When suitable stone for masonry is at 
hand and a suitable subsoil for a foundation is obtainable without 
too much excavation, a masonry abutment will be the best. 
Such an abutment would probably be used when masonry foot- 
ings for trestle bents were employed (§ 139, c). 

Another method is to construct a "crib'' of 10"Xl2" timber. 



§143. 



TKESTLES. 



173 



laid horizontally, drift-bolted together, securely braced and 
embedded into the ground. Except for temporary construction 
such a method is generally 
objectionable on account of 
rapid decay. 

Another method, used most 
commonly for pile trestles, and 
for framed trestles having pile 
foundations (§ 139, a), is to use 
a pile bent at such a place that 
the natural surface on the up^ 
hill side is not far below the 
cap, and the thrust of the material, fiUed in to bring the surface 
to grade, is insignificant. 3"Xl2" planks are placed behind 
the piles, cap, and stringers to retain'the filled material. 




Fig. 79. 



FLOOR SYSTEMS. 

143. Stringers. The general practice is to use two, three, 
and even four stringers under each rail. Sometimes a stringer 
is placed under each guard-rail. Generally the stringers are 
made of two panel lengths and laid so that the joints alternate. 
A few roads use stringers of only one panel length, but this prac- 
tice is strongly condemned by many engineers. The stringers 
should be separated to allow a circulation of air around them 
and prevent the decay which would occur if they were placed 
close together. This is sometimes done by means of 2'' planks, 
6' to 8' long, which are placed over each trestle bent. Several 
bolts, passing through all the stringers forming a group and 
through the separators, bind them all into one solid construc- 
tion. Cast-iron ^' spools'^ or washers, varying from 4'' to J" 
in length (or thickness), are sometimes strung on each bolt so 
as to separate the stringers. Sometimes washers are used 
between the separating planks and the stringers, the object of 
the separating planks then being to bind the stringers, especially 
abutting stringers, and increase their stiffness. 

The most coramon size for stringers is 8''Xl6". The Penn- 
sylvania Railroad varies the width, depth, and number of 
stringers under each rail according to the clear span. It may 
be noticed that, assuming a uniform load per running foot, both 
the pressure per square inch at the ends of the stringers (the 



174 



RAILROAD CONSTRUCTION. 



§ 144. 



caps having a width of 12") and also the stress due to trans- 
verse strain are kept approximately constant for the variable 
gross load on these varying spans. 



Clear span. 


No. of pieces 
under each rail. 


Width. 


Depth. 


10 feet 
12 ♦• 
14 ** 


2 
2 
3 


8 inches 

10 ♦• 

10 ** 


16 inches 
16 " 
16 ** 



144. Corbels. A corbel (in trestle-work) is a stick of timber 
(perhaps two placed side by side), about 3' to 6' long, placed 
underneath and along the stringers and resting on the cap. 
There are strong prejudices for and against their use, and a 
corresponding diversity in practice. They are bolted to the 
stringers and thus stiffen the joint. They certainly reduce the 
objectionable crushing of the fibers at each end of the stringer, 
but if the corbel is no wider than the stringers, as is generally 
the case, the area of pressure between the corbels and the cap is 




Fig. 80. 

no greater and the pressure per square inch on the cap is no less 
than the pressure on the cap if no corbels were used. If the 
corbels and cap are made of hard wood, as is recommended by 
some, the danger of crushing is lessened, but the extra cost and 
the frequent scarcity of hard wood, and also the extra cost and 
labor of using corbels, may often neutralize the advantages 
obtained by their use. 

145. Guard-rails. These are frequently made of 5''X8" stuff, 
notched 1'' for each tie. The sizes vary up to 8"X8", and the 
depth of notch from J" to IJ". They are generally bolted to 
every third or fourth tie. It is frequently specified that they 
shall be made of oak, white pine, or yellow pine. The joints 
are made over a tie, by halving each piece, as illustrated in Fig. 
81. The joints on opposite sides of the trestle should be "stag- 



§ 146. TRESTLES, 175 

gered." Some roads fasten every tie to the guard-rail, using a 
bolt, a spike, or a lag-screw. 

Guard-rails were originally used with the idea of preventing 
the wheels of a derailed truck from running off the ends of the 
ties. But it has been foimd that an outer guard-rail alone (with- 
out an inner guard-rail) becomes an actual element of danger, 
since it has frequently happened that a derailed wheel has caught 
on the outer guard-rail, thus causing the truck to slew aroimd 




Fig. 81. 

and so produce a dangerous accident. The true function of the 
outside guard-rail is thus changed to that of a tie-spacer, which 
keeps the ties from spreading when a derailment occurs. The 
inside guard-rail generally consists of an ordinary steel rail 
spiked about 10 inches inside of the running rail. These inner 
guard-rails should be bent inward to a point in the center of the 
track about 50 feet beyond the end of the bridge or trestle. If 
the inner guard-rails are placed with a clear space of 10 inches 
inside the running rail, the outer guard-rails should be a^ least 
6' 10'' apart. They are generally much farther apart than this. 

146. Ties on trestles. If a car is derailed on a bridge or 
trestle, the heavily loaded wheels are apt to force their way be- 
tween the ties by displacing them unless the ties are closely 
spaced and fastened. The clear space between ties is generally 
equal to or less than their width. Occasionally it is a little more 
than their width. 6''X8'' ties, spaced 14'' to 16" from center 
to center, are most frequently used. The length varies from 
9' to 12' for single track. They are generally notched J" deep 
on the under side where they rest on the stringers. Oak ties 
are generally required even when cheaper ties are used on the 
other sections of the road. Usually every third or fourth tie is 
bolted to the stringers. When stringers are placed underneath 
the guard-rails, bolts are run from the top of the guard-rail to 
the under side of the stringer. The guard-rails thus hold down 
the whole system of ties, and no direct fastening of the ties to 
the stringers is needed. 

147. Superelevation of the outer rail on curves. The location 
of curves on trestles should be avoided if possible, especially 
when the trestle is high. Serious additional strains are intro- 



176 



RAILROAD CONSTRUCTION. 



§147. 



duced especially when the curvature is sharp or the speed high. 
Since such curves are sometimes practically unavoidable, it is 
necessary to design the trestle accordingly. If a train is stopped 
on a curved trestle, the action of the train on the trestle is 
evidently vertical. If the train is moving with a considerable 
velocity, the resultant of the weight and the centrifugal action 
is a force somewhat inclined from the vertical. Both of these 
conditions may be expected to exist at times. If the axis of 
the system of posts is vertical (as illustrated in methods a,h,c, d, 
and e), any lateral force, such as would be produced by a mov- 
ing train, will tend to rack the trestle bent. If the stringers are 
set vertically, a centrifugal force likewise tends to tip them 
side wise. If the axis of the system of posts (or of the stringers) 
is inclined so as to coincide with the pressure of the train on the 
trestle when the train is moving at its normal velocity, there is 
no tendency to rack the trestle when the train is moving at that 
velocity, but there will be a tendency to rack the trestle or 
twist the stringers when the train is stationary. Since a moving 
train is usually the normal condition of affairs, as well as the 
condition which produces the maximum stress, an inclined axis 
is evidently preferable from a theoretical standpoint; but what- 
ever design is adopted, the trestle should evidently be suffi- 
ciently cross-braced for either a moving or a stationary load, 
and any proposed design must be studied as to the effect of both 
of these conditions. Some of the various methods of securing 
the requisite superelevation may be described as follows : 

(a) Framing the outer posts longer than the inner posts, so 

that the cap is inclined at the 
proper angle; axis of posts verti- 
cal. (Fig. 82.) The method re= 
quires more work in framing the 
trestle, but simplifies subsequent 
track-laying and maintenance, un-= 
less it should be found that the 
superelevation adopted is unsuit- 
able, in which case it could be cor- 
, rected by one of the other methods 
given below. The stringers tend 
to twist when the train is sta- 
tionaiy. 

(b) Notching the cap so that the stringers are at a different 




Fig. 82. 



§147. 



TRESTLES. 



177 



L 



FiQ. 83. 



elevation. (Fig. 83.) This weakens the cap and requires that 
all ties shall be notched to a 
bevelled surface to fit the string- 
ers, which also weakens the ties. 
A centrifugal force will tend to 
twist the stringers and rack the 
trestle. 

(c) Placing wedges underneath 
the ties at each stringer. These 
wedges are fastened Tvdth two 
bolts. Two or more wedges will 
be required for each tie. The ad- 
ditional number of pieces required 
for a long curve will be immense, and the work of inspection and 
keeping the nuts tight Tvdll greatly increase the cost of main- 
tenance. 

(d) Placing a wedge under the outer rail at each tie. This 
requires but one extra piece per tie. There is no need of a 
wedge under the inner tie in order to make the rail normal to 
the tread. The resulting inward inclination is substantially that 
produced by some forms of rail-chairs or tie-plates. The spikes 
(a Httle longer than usual) are driven through the wedge into 
the tie. Sometimes ^'lag-screws'' are used instead of spikes. 
If experience proves that the superelevation is too much or too 
little, it may be changed by this method with less work than 
by any other. 

(e) Corbels of different heights. When corbels are used (see 

§ 144) the required in- 
cUnation of the floor sys- 
tem may be obtained by 
varying the depth of the 
corbels. 

(f) Tipping the whole 
trestle. This is done by 
placing the trestle on an 
inclined foundation. If 
very much inclined, the 
trestle bent must be se- 
cured against the possi- 
bihty of slipping sidewise, 
for the slope would be considerable with a sharp curve, and the 




Fig. 84. 



178 KAILROAD CONSTRUCTION. § 148. 

vibration of a moving train would reduce the coefficient o! 
friction to a comparatively small quantity. 

(g) Framing the outer posts longer. This case is identical 
with case (a) except that the axis of the system of posts is 
inclined, as in case (/), but the sill is horizontal. 

The above-described plans will suggest a great variety of 
methods which are possible and which differ from the above 
only in minor details. 

148. Protection from fire. Trestles are peculiarly subject to 
fire, from passing locomotives, which may not only destroy the 
trestle, but perhaps cause a terrible disaster. This danger is 
sometimes reduced by placing a strip of galvanized iron along 
the top of each set of stringers and also along the tops of the 
caps. Still greater protection was given on a long trestle on the 
Louisville and Nashville R. R. by making a solid flooring of 
timber, covered with a layer of ballast on which the ties and 
rails were laid as usual. 

Barrels of water should be provided and kept near all trestles, 
and on very long trestles barrels of water should be placed every 
two or three hundred feet along its length. A place for the bar- 
rels may be provided by using a few ties which have an extra 
length of about four feet, thus forming a small platform, which 
should be surrounded by a railing. The track-walker should be 
held accountable for the maintenance of a supply of water in 
these barrels, renewals being frequently necessary on account of 
evaporation. Such platforms should also be provided as refuge- 
bays for track-walkers and trackmen working on the trestle. On 
very long trestles such a platform is sometimes provided with 
sufficient capacity for a hand-car. 

149. Timber. Any strong durable timber may be used when 
the choice is limited, but oak, pine, or cypress are preferred 
when obtainable. When all of these are readily obtainable, 
the various parts of the trestle will be constructed of different 
kinds of wood — the stringers of long-leaf pine, the posts and 
braces of pine or red cypress, and the caps, sills, and corbels (if 
used) of white oak. The use of oak (or a similar hard wood) 
for caps, sills, and corbels is desirable because of its greater 
strength in resisting crushing across the grain, which is the 
critical test for these parts. There is no physiological basis to 
the objection, sometimes made, that different species of timber, 
in Qontact with each other^ will rot quicker than if only one 



Oxp;^ 



PLATE IL 




(To face -page 178.) 



§ 150. TRESTLES. 179 

kind of timber is used. When a very extensive trestle is to be 
built at a place where suitable growing timber is at hand but 
there is no convenient sawmill, it will pay to transport a port- 
able sawmill and engine and cut up the timber as desired. 

150. Cost of framed timber trestles. The cost varies widely 

on account of the great variation in the cost of timber. When 

a railroad is first penetrating a new and imdeveloped region, the 

cost of timber is frequently small, and when it is obtainable from 

the company's right-of-way the only expense is feUing and 

sawing. The work per M, B. M., is small, considering that a 

single stick 12'' X 12'' X 25' contains 300 feet, B. M., and that 

sometimes a few hours' work, worth less than $1, will finish all 

the w^ork required on it. Smaller pieces wiU of course require 

more work per foot, B. M. Long-leaf pine can be purchased 

from the mills at from $S to $12 per M feet, B. M., according 

to the dimensions. To this must be added the freight and labor 

: of erection. The cartage from the nearest railroad to the trestle 

, may often be a considerable item. Wrought iron will cost 

; about 3 c. per pound and cast iron 2 c, although the prices are 

often lower than these. The amount of iron used depends on 

' the detailed design, but, as an average, will amount to $1.50 

I to $2 per 1000 feet, B. M., of timber. A large part of the tres- 

I tling of the country has been built at a contract price of about 

' $30 per 1000 feet, B. M., erected. While the cost will frequently 

rise to $40 and even $50 when timber is scarce, it will drop to 

$13 (cost quoted) when timber is cheap. 

I 

I DESIGN OF WOODEN TRESTLES. 

! 151. Common practice. A great deal of trestling has been 
j constructed without any rational design except that custom and . 

experience have shown that certain sizes and designs are probably 
{ safe. Tliis method has resulted occasionally in failures but more 
, frequently in a very large waste of timber. Many railroads 
I employ a uniform size for all posts, caps, and sills, and a uniform 
1 size for stringers, all regardless of the height or span of the 

trestle. For repair work there are practical reasons favoring 
j this. ^'To attempt to run a large lot of sizes would be more 
I wasteful in the end than to maintain a few stock sizes only. 

Lumber can be bought more cheaply by giving a general order 

for ' the run of the mill for the season/ or ' sl cargo lot/ specif j^ 



180 RAILROAD CONSTRUCTION. § 152. 

ing approximate percentages of standard stringer size, of 
12 X 12-inch stuff, 10 X 10-inch stuff, etc., and a Hberal propor- 
tion of 3- or 4-inch plank, all lengths thrown in. The 12 X 12- 
inch stuff, etc., is ordered all lengths, from a certain specified 
length up. In case of a wreck, washout, burn-out, or sudden 
call for a trestle to be completed in a stated time, it is much 
more economical and practical to order a certain number of 
carloads of Hrestle stuff' to the ground and there to select piece 
after piece as fast as needed, dependent only upon the length of 
stick required. When there is time to make the necessary sur- 
veys of the ground and calculations of strength, and to wait for a 
special bill of timber to be cut and delivered, the use of differ- 
ent sizes for posts in a structure would be warranted to a certain 
extent." * For new construction, when there is generally 
sufficient time to design and order the proper sizes, such waste- 
fulness is less excusable, and under any conditions it is both 
safer and more economical to prepare standard designs which 
can be made applicable to varying conditions and which will at 
the same time utilize as much of the strength of the timber as 
can be depended on. In the following sections will be given 
the elements of the preparation of such standard designs, which 
will utilize uniform sizes with as little waste of timber as possible. 
It is not to be understood that special designs should be made 
for each individual trestle. 

152. Required elements of strength. The stringers of trestles 
are subject to transverse strains, to crushing across the grain 
at the ends, and to shearing along the neutral axis. The strength 
of the timber must therefore be computed for all these kinds 
of stress. Ca'ps and sills will fail, if at all, by crushing across 
the grain; although subject to other forms of stress, these could 
hardly cause failure in the sizes usually employed. There is an 
apparent exception to this: if piles are improperly driven and 
an uneven settlement subsequently occurs, it may have the 
effect of transferring practically all of the weight to two or three 
piles, while the cajp is subjected to a severe transverse strain 
which may cause its failure. Since such action is caused gener- 
ally by avoidable errors of construction it may be considered as 
abnormal, and since such a failure will generally occur by a 
gradual settlement, all danger may be avoided by reasonable 

♦ From "Economical Designing of Timber Trestle Bridges." 



§ 153. TRESTLES. 181 

care in inspection. Posts must be tested for their columnar 
strength. These parts form the bulk of the trestle and are the 
parts which can be definitely designed from known stresses. 
The stresses in the bracing are more indefinite, depending on 
indeterminate forces, since the inclined posts take up an un- 
known proportion of the lateral stresses, and the design of the 
bracing may be left to what experience has shown to be safe, 
without invohdng any large waste of timber. 

153. Strength of timber. Until recently tests of the strength 
of timber have generally been made by testing small, selected, 
well-seasoned sticks of "clear stuff," free from knots or imper- 
fections. Such tests would give results so much higher than 
the vaguely known strength of large unseasoned " comLmerciaP' 
timber that very large factors of safety were recommended — 
factors so large as to detract from any confidence in the whole 
theoretical design. Recently the U. S. Government has been 
making a thoroughly scientific test of the strength of full-size 
timber under various conditions as to seasoning, etc. The work 
has been so extensive and thorough as to render possible the 
economical designing of timber structures. 

One important result of the investigation is the determina- 
tion of the great influence of jthe moisture in the timber and 
the law of its effect on the strength. It has been also shown 
that timber soaked vnth water has substantially the same 
strength as green timber, even though the tirxiber had once been 
thoroughly seasoned. Since trestles are exposed to the weather 
they should be designed on the basis of using green timber. 
It has been showTi that the strength of green timber is very 
regularly about 55 to 60% of the strength of timber in which 
the moisture is 12% of the dry weight, 12% being the proportion 
of moisture usually found in timber that is protected from the 
weather but not heated, as, e.g., the timber in a barn. Since 
the moduH of ruptiu-e have all been reduced to this standard of 
moisture (12%), if we take one-eighth of the rupture values, it 
still allows a factor of safety of about five, even on green timber. 
On page 182 there are quoted the values taken from the U. S- 
Government reports on the strength of timber, the tests 
probably being the most thorough and reHable that were ever 
made. 

On page 183 are given the ''working unit stresses for^structural 
timber, expressed in pounds per square inch," as recommended by 



182 



KAILROAD CONSTRUCTION. 



§ 151 



the committee on "Wooden Bridges and Trestles," of the Amer- 
ican Railway Engineering Association. The report was pre- 
sented at their tenth annual convention, held in Chicago, in 
March, 1909. 

Moduli of rupture for various timbers. [12% moisture.] 
(Condensed from U. S. Forestry Circular, No. 15.) 











Cross-bending. 


Crush- 
ing 
end- 
wise. 


^ CD 

r 


m 


No. 


Species. 


^1 
11 


Modulus 

of 
Elasticity. 


1 
2 
3 
4 
5 
6 
7 


Long-leaf pine .... 
Cuban " .... 

Short-leaf *' 

Loblolly '• .... 

White '• 

Red " .... 
Spruce ** .... 


38 
39 
32 
33 
24 
31 
39 


12 600 

13 600 
10 100 
11300 

7 900 

9 100 

10 000 


2 070 000 
2 370 000 

1 680 000 

2 050 000 
1 390 000 
1 620 000 
1 640 000 


8000 
8700 
6500 
7400 
5400 
6700 
7300 


1180 
1220 

960 
1150 

700 
1000 
1200 


700 
700 
700 
700 
400 
500 
800 


8 

9 

10 


Bald cypress . . . 


29 
23 
32 


7 900 

6 300 

7 900 


1 290 000 

910 000 

1 680 000 


6000 
5200 
5700 


800 
700 
800 


500 


White cedar 

Douglas soruce. . . . 


400 
500 


11 


White oak 


50 
46 
50 
46 
45 
46 
45 
46 


13 100 
11300 

12 300 
11500 
11400 

13 100 
10 400 
12 000 


2 090 000 

1 620 000 

2 030 000 
1610 000 
1 970 000 
1 860 000 
1 750 000 
1 930 000 


8500 
7300 
7100 
7400 
7200 
8100 
7200 
7700 


2200 
1900 
3000 
1900 
2300 
2000 
1600 
1800 


1000 


12 


Overcup ** . . 




1000 


13 


Post " . . 




1100 


14 


Cow " . . 




900 


15 


Red *• .. 




1100 


16 


Texan ** . . 




900 


19 


Willow " . . 




900 


?0 


Spanish " . . 




900 








21 
27 
•^8 


Shagbark hickory. . 
Pignut *' 
White elm 


51 
56 
34 
46 
39 


16 000 
18 700 
10 300 
13500 
10 800 


2 390 000 
2 730 000 
1 540 000 
1 700 000 
1 640 000 


9500 
10900 
6500 
8000 
7200 


2700 
3200 
1200 
2100 
1900 


1100 

1200 

800 


?9 


Cedar " 


1300 


30 


White ash 


1100 











154. Loading. As shown in § 138, the span of trestles is always 
small, is generally 14 feet, and is never greater than 18 feet 
except when supported by knee-braces. The greatest load that 
will ever come on any one span will be the concentrated loading 
of the drivers of a consolidation locomotive. With spans of 14 
feet or less it is impossible for even the four pairs of drivers to 
be on the same span at once. The weight of the rails, ties, and 
guard-rails should be added to obtain the total load on the string- 
ers, and the weight of these, plus the weight of the stringers, 
should be added to obtain the pressure on the caps or corbels. 



§ 154. 



TEBSTLES. 



183 



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184 



HAILEOAD CONSTEUCTION. 



§155. 



This dead load is almost insignificant compared with the live 
load and may be included with it. The weight of rails, ties, 
etc., may be estimated at 240 pounds per foot. To obtain the 
weight on the caps the weight of the stringers must be added, 
which depends on the design and on the weight per cubic foot 
of the wood employed. But as the weight of the stringers is 
comparatively small, a considerable percentage of variation in 
weight will have but an insignificant effect on the result. Dis- 
regarding all refinements as to actual dimensions, the ordinary 
maximum loading for standard-gauge railroads may be taken 
as that due to four driving-axles, spaced 5' 0'' apart and giving 
a pressure of 40000 pounds per axle. This should be increased 
to 54000 pounds per axle (same spacing) for the heaviest traffic. 
On the basis of 40000 pounds per axle or 20000 pounds per wheel 
the following results have been computed: This loading is 
assumed to allow for impact. 

STRESSES ON VARIOUS SPANS DUE TO MOVING LOADS OF 20000 
POUNDS, SPACED 5' 0'' APART, WITH 120 POUNDS PER FOOT 
OF LIVE LOAD. 



Span in feet. 


Max. moment, 
ft. lbs. 


Max shear. 


Max. load on 

one cap under 

one rail. 


10 
12 
14 
16 
18 


51 500 

82 160 

112 940 

123 840 

164 860 


30 600 
35 720 
39 410 
43 460 

47 747 


41 200 
49 440 
57 680 
65 920 
75 160 



Although the dead load does not vary in proportion to the 
live load, yet, considering the very small influence of the dead 
load, there will be no appreciable error in assuming the corre- 
sponding values, for a load of 54000 lbs. per axle, to be ^^ of 
those given in the above tabulation. 

155. Factors of safety. The most valuable result of the gov- 
ernment tests is the knowledge that under given moisture condi- 
tions the strength of various species of sound timber is not the 
variable uncertain quantity it was once supposed to be, but that 
its strength can be relied on to a comparatively close percentage. 
This confidence in values permits the employment of lower fac- 
tors of safety than have heretofore been permissible. Stresses, 
which when excessive would result in immediate destruction, 
such as cross-breaking and columnar stresses, should be allowed 
a higher factor of safety— say 6 or 8 for green timber. Other 
stresses, such as crushing across the grain and shearing along the 



§ 156. TRESTLES. 1S5 

neutral axis, which will be apparent to inspection before it is 
dangerous, may be allowed lower factors — say 3 to 5. 

156. Design of stringers. The strength of rectangular beams 
of equal width varies as the square of the depth; therefore deep 
beams are the strongest. On the other hand, when any cross- 
sectional dimension of timber much exceeds 12^' the cost is 
much higher per M, B. M., and it is correspondingly difficult to 
obtain thoroughly sound sticks, free from wind-shakes, etc. 
Wind-shakes especially affect the shearing strength. Also, if 
the required transverse strength is obtained by using high nar- 
row stringers, the area of pressure between the stringers and the 
cap may become so small as to induce crushing across the grain. 
This is a very common defect in trestle design. As already in- 
dicated in § 138, the span should vary roughly with the average 
height of the trestle, the longer spans being employed when the 
trestle bents are very high, although it is usual to employ the 
same span throughout any one trestle. 

To illustrate, if we select a span of 14 feet, the load on one 
cap will be 57680 lbs. If the stringers and cap are made of 
long-leaf yellow pine, the allowable value, according to the table 
on page 183 for "compression across the grain'' is 260 pounds 
per square inch; this will require 222 square inches of surface. 
If the cap is 12" wide, this will require a width of 18.5 inches, or 
say 2 stringers under each rail, each 9 inches wide. For rectan- 
gular beams 

Moment = ^R'bh\ 

Using for R' the safe value 1300 lbs. per square inch, we have 

112940X12 = J X1300Xl8X/i2, 

from which h = 18''.7. If desired, the width may be increased 
to 10" and the depth correspondingly reduced, which will give 
similarly h = W.7 or say 18". This shows that two beams, 
10"X18", under each rail will stand the transverse bending and 
have more than enough area for crushing. 
The shear per square inch will equal 

3 total shear 3 39410 ,^, „ 

2 cross-section ==2 2X10X18 ^ ^^^ ^^^' P^^ ^^- "^^^• 

This is higher than the recommended working value. The com- 
bination suggested on page 174, viz., 3 beams 10"X16" for 14 feet 
span, gives a far safer value. Considering that wooden beams, 



186 BAILHOAD OONSTRUCTlONo § 157. 

tested to destruction, usually fail by shearing, the three-beam 
combination is safer. 

The deflection should be computed to see if it exceeds the 
somewhat arbitrary standard of -^J^ of the span. The deflec- 
tion for uniform loading is 

S2bh^E ' 



in which Z= length in inches; 

TF= total load, assumed as uniform; 

E= modulus of elasticity, given as 850,000 lbs. 

per sq. in. for long-leaf pine, according to the table on p. 183. 
Then 

._ 5X57680X168^ 
32X20X183X850000 

4X168"=0^84, 

so that the calculated deflection is well within the limit. Of 
course the loading is not strictly uniform, but even with a Ub- 
eral allowance the deflection is still safe. 

For the heaviest practice (54000 lbs. per axle) these stringer 
dimensions must be correspondingly increased. 

157. Design of posts. Four posts are generally used for 
single-track work. The inner posts are usually braced by the 
cross-braces, so that their columnar strength is largely increased; 
but as they are apt to get more than their share of work, the ad- 
vantage is compensated and they should be treated as unsup- 
ported columns for the total distance between cap and sill in 
simple bents, or for the height of stories in multiple-story con- 
struction. The caps and sills are assumed to have a width of 12". 
It facilitates the application of bracing to have the columns of 
the same width and vary the other dimension as required. 

Unfortunately the experimental work of the U. S. Govern- 
ment on timber testing has not yet progressed far enough to 
establish unquestionably a general relation between the strength 
of long columns and the crushing strength of short blocks. The 



§ 157. TRESTLES. 187 

following formula has been suggested, but it cannot be consid- 
ered as established: 

, ^,, 700 + 15c . , . , 



/= allowable working stress per sq. in for long columns; 
F=^ '' '' '' '' '* '' '' short blocks; 

I 

Z= length of column in inches; 

cZ= least cross-sectional dimensions in inches. 

Enough work has been done to give great reliability to the two 
following formulae for white pine and yellow pine, quoted from 
Johnson's ^'Materials of Construction," p. 684: 

1 /Z\ 2 
Working load per sq. in. =p = 1000 — "j ( 77) j long-leaf pine; 

'' '' '' '^ =2^= 600- i(^|y, white pine; 

in which Z= length of column in inches, and 

/i=least cross-sectional dimension in inches. 

The frequent practice is to use 12" X 12" posts for all trestles. 
If we substitute in the above formula Z =20' =240" and h = Vl" 
we have p = 1000 -J(-V/)2_ 900 lbs. 

900X144 = 129600 lbs., the working load for each post. This 
is more than the total load on one trestle bent and illustrates 
the usual great waste of timber. Making the post 8" X 12" and 
calculating similarly, we have p = 775, and the working load per 
column is 775X96=74400 lbs. As considerable must be 
allowed for ^^ weathering," which destroys the strength of the 
outer layers of the w^ood, and also for the dynamic effect of 
the live load, 8" X 12" may not be too great, but it is certainly 
a safe dimension. 12" X 6" would possibly prove amply safe 
in practice. One method of allowing for weathering is to dis- 
regard the outer half-inch on all sides of the post, i.e., to cal- 
culate the strength of a post one inch smaller in each dimension 
than the post actually employed. On this basis an 8" X 12" X 20' 



188 KAILROAD CONSTRUCTION. § 158. 

post, computed as a 7"Xll' post, would have a safe columnar 
strength of 706 lbs. per square inch. With an area of 77 square 
inches, this gives a working load of 54362 lbs. for each post, or 
217448 lbs. for the four posts. Considering that 74200 lbs. 
is the maximum load on one cap (14 feet span), the great excess 
of strength is apparent. 
Utilizing the formula given on page 183 for long-leaf pine, 

we have safe stress = 1300 ( 1 —aTyR ) ? in which L = 20 ft. =240 in., 

and Z) = 12 ins. Then safe stress = 1300(1 -i) =867 lbs. per 
square inch. This is practically a check on the above calcula- 
tions. 

158. Design of caps and sills. The stresses in caps and sills 
are very indefinite, except as to crushing across the grain. As 
the stringers are placed almost directly over the inner posts, and 
as the sills are supported just under the posts, the transverse 
stresses are almost insignificant. In the above case four posts 
have an area of 4 X 12" X 8'' =384 sq. in. The total load, 
74200 lbs. 5 will then give a pressure of 193 pounds per square 
inch, which is within the allowable limit. This one feature 
might require the use of 8''Xl2'' posts rather than 6''Xl2" 
posts, for the smaller posts, although probably strong enough as 
posts, would produce an objectionably high pressure. 

159. Bracing. Although some idea of the stresses in the 
bracing could be found from certain assumptions as to wind- 
pressure, etc., yet it would probably not be found wise to de- 
crease, for the sake of economy, the dimensions which practice 
has shown to be sufiicient for the work. The economy that 
would be possible would be too insignificant to justify any risk. 
Therefore the usual dimensions, given in §§ 139 and 140, should 
be employed. 



CHAPTER V. 
TUNNELS. 
SURVEYING^ 

1 60. Surface surveys. As tunnels are always dug from each 
end and frequently from one or more intermediate shafts, it is 
necessary that an accurate surface survey should be made 
between the two ends. As the natural surface in a locality 
where a tunnel is necessary is almost invariably very steep and 
rough, it requires the employment of unusually refined methods 
of work to avoid inaccuracies. It is usual to run a line on the 
surface that will be at every point vertically over the center line 
of the tunnel. Tunnels are generally made straight unless 
curves are absolutely necessary, as curves add greatly to the 
cost. Fig. 85 represents roughly a longitudinal section of the 




0---*i ^00-— *i 6000- 1 -5000" 

Fia. 85. — Sketch of Section of the Hoosac Tunnel, 



Hoosac Tunnel. Permanent stations were located at A, B, C, 
D, E, and F, and stone houses were built at A, B, C, and D. 
These were located with ordinary field transits at first, and then 
all the points were placed as nearly as possible in one vertical 
plane by repeated trials and minute corrections, using a very 
large specially constructed transit. The stations D and F were 
necessary because E and A were invisible from C and B. The 
alignment at A and E having been determined with great accu- 
racy, the true aHgnment was easily carried into the tunnel. 

189 



190 RAILROAD CONSTRUCTION. § 161. 

The relative elevations of A and E were determined with 
great accuracy. Steep slopes render necessary many settings 
of the level per unit of horizontal distance and require that the 
work be unusually accurate to obtain even fair accuracy per 
unit of distance. The levels are usually re -run many times 
until the probable error is a very small quantity 

The exact horizontal distance between the two ends of the 
tunnel must also be known, especially if the tunnel is on a 
grade. The usual steep slopes and rough topography likewise 
lender accurate horizontal measurements very difficult. Fre- 
quently when the slope is steep the measurement is best ob- 
tained by measuring along the slope and allowing for grade. 
This may be very accurately done by employing two tripods 
(level or transit tripods serve the purpose very well), setting 
them up slightly less than one tape-length apart and measuring 
between horizontal needles set in wooden blocks inserted in the 
top of each tripod. The elevation of each needle is also observed. 
The true horizontal distance between two successive positions 
of the needles then equals the square root of the difference of 
the squares of the inclined distance and the difference of eleva- 
tion. Such measurements will probably be more accurate than 
those made by attempting to hold the tape horizontal and 
plumbing down with plumb-bobs, because (1) it is practically 
difficult to hold both ends of the tape truly horizontal; (2) on 
steep slopes it is impossible to hold the down-hill end of a 100- 
foot tape (or even a 25-foot length) on a level with the other 
end, and the great increase in the number of applications of the 
unit of measurement very greatly increases the probable error 
of the whole measurement; (3) the vibrations of a plumb-bob 
introduce a large probability of error in transferring the meas- 
urement from the elevated end of the tape to the ground, and 
the increased number of such applications of the unit of meas- 
urement still further increases the probable error. 

i6i. Surveying down a shaft. If a shaft is sunk, as at ^S, 
Fig 85, and it is desired to dig out the tunnel in both directions 
from the foot of the shaft so as to meet the headings from the 
outside, it is necessary to know, when at the bottom of the 
shaft, the elevation, alignment, and horizontal distance from 
each end of the tunnel. 

The elevation is generally carried down a shaft by means of 
a steel tape. This method involves the least number of appli- 



§ 161. TUNNELS. 191 

cations of the unit of measurement and greatly increases the 
accuracy of the final result. 

The horizontal distance from each end ma.y be easily trans- 
ferred down the shaft by means of a plumb -bob, using some of 
the precautions described in the next paragraph. 

To transfer the alignment from the surface to the bottom of 
a shaft requires the highest skill because the shaft is always 
small; and to produce a line perhaps several thousand feet long 
in a direction given by two points 6 or 8 feet apart requires 
that the two points must be determined with extreme accuracy. 
The eminently successful method adopted in the Hoosac Tunnel 
will be briefly described: Tw^o beams were securely fastened 
across the top of the shaft (1030 feet deep), the beams being 
placed transversely to the direction of the tunnel and as far 
apart as possible and yet allow plumb-lines, hung from the 
intersection of each beam with the tunnel center line, to swdng 
freely at the bottom of the shaft. These intersections of the 
beams wdth the center line were determined by averaging the 
results of a large number of careful observations for alignment. 
Two fine parallel wdres, spaced about yV apart, w^ere then 
stretched between the beams so that the center line of the 
tunnel bisected at all points the space betw^een the wires. 
Plumb-bobs, w^eighing 15 pounds, w^ere suspended by fine wires 
beside each cross-beam, the wires passing between the two 
parallel alignment wdres and bisecting the space. The plumb- 
bobs w^ere allow- ed to swdng in pails of water at the bottom. 
Drafts of air up the shaft required the construction of boxes 
surrounding the wires. Even these precautions did not suffice 
to absolutely prevent vibration of the wire at the bottom 
through a very small arc. The mean point of these vibrations 
in each case was then located on a rigid cross-beam suitably 
placed at the bottom of the shaft and at about the level of the 
roof of the tunnel. Short plumb-lines were then suspended 
from these points whenever desired; a transit was set (by trial) 
so that its line of collimation passed through both plumb-lines 
and the line at the bottom could thus be prolonged. 

Some recent experience in the ^'Tamarack" shaft, 4250 feet 
deep, shows that the accuracy of the results may be affected by 
air-currents to an unsuspected extent. Two 50-lb. cast-iron 
plumb-bobs w^ere suspended wdth No. 24 piano-wdre in this 
shaft. The carefully measured distances between the wires 



192 



BAILROAD CONSTRUCTION. 



162. 



at top and bottom were 16.32 and 16.43 feet respectively. 
After considerable experimenting to determine the cause of 
the variation, it was finally concluded that air-currents were 
alone responsible. The variation of the bobs from a true ver- 
tical plane passing through the wires at the top was of course 
an unknown quantity, but since the variation in one direction 
amounted to 0.11 foot, the accuracy in other directions was 
very questionable. This shows that a careful comparative 
measurement between the wires at top and bottom should 
always be made as a test of their parallelism. 

162. Underground surveys. Survey marks are frequently 
placed on the timbering, but they are apt to prove unreliable 
on account of the shifting of the timbering due to settlement 
of the surrounding material. They should never be placed at 
the bottom of the tunnel on account of the danger of being 
disturbed or covered up. Frequently holes are drilled in the 
roof and filled with wooden plugs in which a hook is screwed 
exactly on line Although this is probably the safest method, 
even these plugs are not always undisturbed, as the material, 
unless very hard, will often settle slightly as the excavation 
proceeds. When a tunnel is perfectly straight and not too long, 
alignment-points may be given as frequently as desired from 

permanent stations located outside 
the tunnel where they are not liable 
to disturbance. This has been ac- 
complished by running the align- 
ment through the upper part of the 
cross-section, at one side of the cen- 
ter, where it is out of the way of 
the piles of masonry material, 
debris, etc., which are so apt to 
choke up the lower part of the 
cross-section. The position of this 
line relative to the cross-section 
being fixed, the alignment of any 
required point of the cross-section 
is readily found by means of a light 
frame or template with a fixed tar- 
get located where this line would intersect the frame when 
properly placed. A level-bubble on the frame will assist in 
setting the frr.me in its proper position. 




FiQ. 86. 



§ 163. TUNNELS. 193 

In all tunnel surveying the cross-mres must be illuminated 
by a lantern, and the object sighted at must also be illuminated. 
A powerful dark-lantern with the opening covered with ground 
glass has been found useful. This may be used to illuminate a 
plumb-bob string or a very fine rod, or to place behind a brass 
plate having a narrow slit in it, the axis of the slit and plate 
being coincident with the plumb-bob string by which it is 
hung. 

On account of the interference to the surveying caused by 
the work of construction and also by the smoke and dust in the 
air resulting from the blasting, it is generally necessary to make 
the surveys at times when construction is temporarily sus- 
pended. 

163. Accuracy of tunnel surveying. Apart from the very 
natural desire to do surveying which shall check well, there is 
an important financial side to accurate tunnel surveying. If 
the survey lines do not meet as desired when the headings come 
together, it may be found necessar}^, if the error is of appreciable 
size, to introduce a slight curve, perhaps even a reversed curve, 
into the alignment, and it is even conceivable that the tiumel 
section would need to be enlarged somewhat to allow for these 
curves. The cost of these changes and the perpetual anno^^ance 
due to an enforced and undesirable alteration of the original 
design will justify a considerable increase in the expenses of the 
survey. Considering that the cost of surveys is usually but a 
small fraction of the total cost of the work, an increase of 10 or 
even 20% in the cost of the surveys will mean an insignificant 
addition to the total cost and frequent'/, if not generally, it will 
result in a sa^dng of many times the increased cost. The 
accuracy actually attained in two noted American tunnels is 
given as follows: The Musconetcong tunnel is about 5000 feet 
long, bored through a mountain 400 feet high. The error of 
alignment at the meeting of the headings was 0^04, error of 
levels 0'.015, error of distance 0'.52. The Hoosac tunnel is 
over 25000 feet long. The heading from the east end met the 
heading from the central shaft at a point 11274 feet from the 
east end and 1563 feet from the shaft. The error in alignment 
was j% of an inch, that of levels ^'a few hundredths," error of 
distance ^'trifling.'' The alignment, corrected at the shaft, 
was carried on through and met the heading from the west end 
at a point 10138 feet from the west end and 2056 feet from 



194 KAILROAD CONSTRUCTION. § 164. 

the shaft. Here the error of alignment was -/g^' and that of 
levels 0.134 ft. 

DESIGN. 

164. Cross-sections. Nearly all tunnels have cross-sections 
peculiar to themselves — all varying at least in the details. The 
general form of a great many tunnels is that of a rectangle sur- 
mounted by a semi-circle or semi-ellipse. In very soft material 
an inverted arch is necessary along the bottom. In such cases 
the sides will generally be arched instead of vertical. The sides 
are frequently battered. With very long tunnels, several forms 
of cross-section will often be used in the same tunnel, owing to 
differences in the material encountered. In solid rock, which 
will not disintegrate upon exposure, no lining is required, and 
the cross-section will be the irregular section left by the blasting, 
the only requirement being that no rock shall be left within the 
required cross-sectional figure. Farther on, in the same tunnel, 
when passing through some very soft treacherous material, it 
may be necessary to put in a full arch lining — top, sides, and 
bottom — which will be nearly circular in cross-section. For 
an illustration of this see Figs. 87 and SS. 

The width of tunnels varies as greatly as the designs. Single- 
track tunnels generally have a width of 15 to 16 feet. Occa- 
sionally they have been built 14 feet wide, and even less, and 
also up to 18 feet, especially when on curves. 24 to 26 feet is 
the most common width for double track. Many double-track 
tunnels are only 22 feet wide, and some are 28 feet wide. The 
heights are generally 19 feet for single track and 20 to 22 feet 
for double track. The variations from these figures are con- 
siderable. The lower limits depend on the cross-section of the 
rolling stock, with an indefinite allowance for clearance and ven- 
tilation. Cross-sections which coincide too closely with what is 
absolutely required for clearance are objectionable, because any 
slight settlement of the lining which would otherwise be harm- 
less would then become troublesome and even dangerous. Figs. 
87, 88, and 89 * show some typical cross-sections. 

165. Grade. A grade of at least 0.2% is needed for drainage. 
If the tunnel is at the summit of two grades, the tunnel grade 
should be practically level, with an allowance for drainage, the 

* Drinker's "Tunneling." 



165. 



TUNNELS. 



195 




Fig. 87. — Hoosac Tunnel. Section through Solid Rock. 




Fig. 88. — Hoosac Tunnes. Section through Soft Ground. 



196 



RAILROAD CONSTRUCTION. 



§ 166. 



actual summit being perhaps in the center so as to drain both 
ways. When the tunnel forms part of a long ascending grade, 
it is advisable to reduce the grade through the tunnel unless the 
tunnel is very short The additional atmospheric resistance and 
the decreased adhesion of the driver wheels on the damp rails in 
a tunnel will cause an engine to work very hard and still more 
rapidly vitiate the atmosphere until the accumulation of poison- 
ous gases becomes a source of actual danger to the engineer and 




St. Cloud Tunnel. 



fireman of the locomotive and of extreme discomfort to the 
passengers. If the nominal ruling grade of the road were 
maintained through a tunnel, the maximum resistance would be 
found in the tunnel. This would probably cause trains to stall 
there, which would be objectionable and perhaps dangerous. 

1 66. Lining. It is a characteristic of many kinds of rock 
and of all earthy material that, although they may be self- 
sustaining when first exposed to the atmosphere, they rapidly 
disintegrate and require that the top and perhaps the sides and 
even the bottom shall be lined to prevent caving in. In this 
country, when timber is cheap, it is occasionally framed as an 
arch and used as the 'permanent lining, but masonry is always 
to be preferred. Frequently the cross-section is made extra 



§167. 



TtJNNfiLS. 



197 



large so that a masonry lining may subsequently be placed inside 
the wooden lining and thus postpone a large expense until the 
road is better able to pay for the work. In very soft unstable 
material, like quicksand, an arch of cut stone voussoirs may be 
necessary to withstand the pressure. A good quality of brick is 
occasionally used for lining, as they are easily handled and make 
good masonry if the pressure is not excessive. Only the best 
of cement mortar should be used, economy in this feature being 
the worst of folly. Of course the excavation must include the 
outside line of the lining. Any excavation which is made out- 
side of this line (by the fall of earth or loose rock or by exces- 
sive blasting) must be refilled with stone well packed in. Occa- 
sionally it is necessary to fill these spaces with concrete. Of 
course it is not necessary that the lining be uniform throughout 
the tunnel. 

167. Shafts. Shafts are variously made with square, rectan- 
gular, elliptical, and circular cross-sections. The rectangular 




Fig. 90. — Connection with Shaft, Church Hill Tunnel. 



cross-section, with the longer axis parallel -with, the tunnel, is 
most usually employed. Generally the shaft is directly over the 
center of the tunnel, but that always impKes a complicated con- 
nection between the linings of the tunnel and shaft, provided 



198 



KAILROAD CONSTRUCTION. 



§168. 



such linings are necessary. It is easier to sink a shaft near to 
one side of the tunnel and make an opening through the nearly 
vertical side of the tunnel. Such a method was employed in the 
Church Hill Tunnel, illustrated in Fig. 90.* Fig. 91 f shows 
a cross-section for a large main shaft. Many shafts have been 
built with the idea of being left open permanently for ventila- 
tion and have therefore been elaborately lined with masonry. 




Fig. 91. — Cross-section. Large Main Shaft. 

The general consensus of opinion now appears to be that shafts 
are worse than useless for ventilation; that the quick passage of 
a train through the tunnel is the most effective ventilator; and 
that shafts only tend to produce cross-currents and are ineffective 
to clear the air. In consequence, many of these elaborately 
lined shafts have been permanently closed, and the more recent 
practice is to close up a shaft as soon as the tunnel is completed. 
Shafts always form drainage -wells for the material they pass 
through, and sometimes to such an extent that it is a serious 
matter to dispose of the water that collects at the bottom, 
requiring the construction of large and expensive drains. 

1 68. Drains. A tunnel will almost invariably strike veins of 
water which will promptly begin to drain into the tunnel and 
not only cause considerable trouble and expense during construc- 
tion, but necessitate the provision of permanent drains for its 
perpetual disposal. These drains must frequently be so large as 



* Drinker's "Tunneling." 

t Rziha, "Lehrbuch der Gesammten Tunnelbaukunst.** 



169, 



TUNNELS, 



199 



to appreciably increase the required cross-section of the tunnel. 
Generally a small open gutter on each side will suffice for this 
purpose, but in double-track tunnels a large covered drain is 
often built between the tracks. It is sometimes necessary to 
thoroughly grout the outside of the lining so that water wiU. not 
force its way through the masonry and perhaps injure it, but 
may freely drain doTMi the sides and pass through openings in 
the side walls near their base into the gutters. 



CONSTRUCTION. 

169. Headings. The methods of all tunnel excavation de- 
pend on the general principle that all earthy material, except 
the softest of liquid mud and quicksand, will be self-sustaining 
over a greater or less area and for a greater or less time after 
excavation is made, and the work consists in excavating some 
material and immediately propping up the exposed surface by 
timbering and poling-boards. The excavation of the cross- 
section begins with cutting out a "heading," which is a small 
horizontal drift whose breast is constantly kept 15 feet or more 
in advance of the full cross-sectional excavation. In solid 
self-sustaining rock, which wiU not decompose upon exposure 
to air, it becomes simpl}^ a matter of excavating the rock with 
the least possible expenditure of time and energy. In soft 
ground the heading must be heavily timbered, and as the heading 
is gradually enlarged the timbering must be gradually extended 
and perhaps replaced, according to some regular system, so that 
when the full cross-section has been ex- 
cavated it is supported by such timbering 
as is intended for it. The heading is 
sometimes made on the center line near 
the top; with other plans, on the center 
line near the bottom; and sometimes two 
simultaneous headings are run in the two 
lower comers. Headings near the bot- 
tom serve the purpose of draining the 
material above it and facilitating the 
excavation. The simplest case of head- 
ing timbering is that shown in Fig. 92, 
in which cross-timbers are placed at in- p^^^ g2. 

tervals just under the roof, set in notches 
cut in the side walls and supporting poling-boards which sus- 




200 



RAILROAD CONSTRUCTION. 



§170. 



tain whatever pressure may come on them. Cross-timbers 
near the bottom support a flooring on which vehicles for trans- 
porting material may be run and under which the drainage 
may freely escape. As the necessity for timbering becomes 
greater, side timbers and even bottom timbers must be added, 
these timbers supporting poling -boards, and even the breast 
of the heading must be protected by boards suitably braced. 




Fig. 93. — Timbering for Tunnel Heading. 



as shown in Fig. 93. The supporting timbers are framed into 
collars in such a manner that added pressure only increases 
their rigidity. 

170. Enlargement. Enlargement is accomplished by remov- 
ing the poling-boards, one at a time, excavating a greater or less 
amount of material, and immediately supporting the exposed 
material with poUug-boards suitably braced. (See Figs. 93 and 
94.) This work being systematically done, space is thereby 
obtained in which the framing for the full cross-section ma}^ be 
gradually introduced. The framing is constructed with a cross- 



§ 171. 



TUNNELS. 



201 



section so large that the masonry lining may be constructed 
within it. 

171. Distinctive features of various methods of construction. 

There are six general systems, known as the English, German, 
Belgian, French, Austrian, and American. They are so named 




Fig. 94. 



from the origin of the methods, although their use is not con- 
fined to the countries named. Fig. 95 shows by numbers (1 to 5) 
the order of the excavation within the cross-sections. The Eng- 
lish, Austrian, and American systems are alike in excavating the 
entire cross-section before beginning the construction of the 
masonry lining. The German method leaves a solid core (5) 
until practically the whole of the lining is complete. This has 
the disadvantage of extremely cramped quarters for work, poor 
ventilation, etc. The Belgian and French methods agree in 
excavating the upper part of the section, building the arch at 
once, and supporting it temporarily until the side walls are 
built. The Belgian method then takes out the core (3), removes 
very short sections of the sides (4) immediately underpinning 
the arch with short sections of the side walls and thus gradually 
constructing the whole side wall. The French method digs out 
the sides (3), supporting the arch temporarily with timbers and 
then replacing the timbers with masonry; the core (4) is taken 
out last. The French method has the same disadvantage as the 
German — working in a cramped space. The Belgian and French 
systems have the disadvantage that the arch, supported tempo- 
rarily on timber, is very apt to be strained and cracked by the 
slight settlement that so frequently occurs in soft material. The 
EngHsh, Austrian, and American methods differ mainly in the 



202 



KAILROAD CONSTRUCTION. 



§ 171. 



design of the timbering. The English support the roof by lines 
of very heavy longitudinal timbers which are supported at com- 
paratively wide intervals by a heavy framework occupying the 





ENGLISH 



AUSTRIAN 





GERMAN 



BELGIAN 




FRENCH 



/ / 




y—- ■ 


' i 


4 


1 3 


1 




Y 


1 1 




1 ^ 



AMERICAN 



Fig. 95. — Order of Working by the Various Systems. 



whole cross-section. The Austrian system uses such frequent 
cross-frames of timber- work that poling-boards will suffice to 
support the material between the frames. The American sys- 
tem agrees with the Austrian in using frequent cross-frames 



§ 172. TUNNELS. 203 

supporting poling-boards, but differs from it in that the " cross- 
frames" consist simply of arches of 3 to 15 wooden voussoirs, 
the voussoirs being blocks of 12''Xl2'' timber about 2 to 8 feet 
long and cut with joints normal to the arch. These arches are 
put together on a centering w^hich is removed as soon as the arch 
is keyed up and thus immediately opens up the full cross-section, 
so that the center core (4) may be immediately dug out and the 
masonry constructed in a large open space. The American sys- 
tem has been used successfully in very soft ground, but its ad- 
vantages are greater in loose rock, w^hen it is much cheaper than 
the other methods which employ more timber. Fig. 90 and 
Plate III illustrate the use of the American system. Fig. 90 
shows the'Wooden arch in place. The masonry arch may be 
placed when convenient, since it is possible to lay the track and 
commence traffic as soon as the wooden arch is in place. The 
student is referred to Drinker's ^^ Tunneling" and to Rziha's 
''Lehrbuch der Gesammten Tunnelbaukunst " for numerous 
illustrations of European methods of tunnel timbering. 

172. Ventilation during construction. Tunnels of any great 
length must be artificially ventilated during construction. If 
the excavated material is rock so that blasting is necessary, the 
need for ventilation becomes still more imperative. The inven- 
tion of compressed-air drills simultaneously solved two difficul- 
ties. It introduced a motive power which is unobjectionable in 
its application (as gas would be), and it also furnished at the same 
time a supply of just what is needed — pure air. If no blasting 
is done (and frequently even when there is blasting), air must be 
supplied by direct pumping. The cooling effect of the sudden 
expansion of compressed air only reduces the otherwise objection- 
ably high temperature sometimes found in tunnels. Since pure 
air is being continually pumped in, the foul air is thereby forced 
out. 

173. Excavation for the portals. Under normal conditions 
there is always a greater or less amount of open cut preceding 
and following a tunnel. Since all tunnel methods depend (to 
some slight degree at least) on the capacity of the exposed ma- 
terial to act as an arch, there is implied a considerable thickness 
of material above the tunnel. This thickness is reduced to 
nearly zero over the tunnel portals and therefore requires special 

treatment, particularly when the material is very soft. Fig. 96 * 

» . — ■ 

* Rziha, "Lehrbuch der Gesammten Timnelbaukunst." 



204 



RAILROAD CONSTRtrCTiON. 



§174, 



illustrates one method of breaking into the ground at a portal. 
The loose stones are piled on the framing to give stability to the 
framing by their weight and also to retain the earth on. the 




Fig 



Timbering for Tunnel Portal. 



^ 



slope above. Another method is to sink a temporary shaft to 
the tunnel near the portal; immediately enlarge to the full size , 
and build the masonry lining ; then work back to the portal J 
This method is more costly, but is preferable in very treacherous 
ground, it being less liable to cause landslides of the surface \ 
material. 

174. Tunnels vs. open cuts. In cases in which an open cut 
rather than a tunnel is a possibility the ultimate consideration 
is generally that of first cost combined with other financial con- 



PLATE III. 




Elevation or Portal. 
Phcbnixville Tdnnkl. p. S. V. R. B 



Longitudinal Section or Pobtal. 



PLATE III. 




Longitudinal Section of Portal. 



§ 175. 



TUNNELS. 



205 



siderations and annual maintenance charges directly or indirectly 
connected with it. Even when an open cut may be constructed 
at the same cost as a tunnel (or perhaps a little cheaper) the 
tunnel may be preferable under the following conditions: 

1. When the soil indicates that the open cut would be liable 
to landslides. 

2. When the open cut would be subject to excessive snow- 
drifts or avalanches. 

3. ^\Tien land is especially costly or it is desired to run under 
existing costly or valuable buildings or monuments. ^Mien run- 
ning through cities, tunnels are sometimes constructed as open 
cuts and then arched over. 

These cases apply to tunnels vs. open cuts when the align- 
ment is fixed by other considerations than the mere topography. 
The broader question of excavating tunnels to avoid excessive 
grades or to save distance or curvature, and similar problems, 
are hardly susceptible of general analysis except as questions of 
railway economics and must be treated individually. 

175. Cost of tunneling. The cost of any construction which 
involves such uncertainties as tunneling is very variable. It 
depends on the material encountered, the amount and kind of 
timbering required, on the size of the cross-section, on the price 
of labor, and especially on the reconstruction that may be neces- 
sary on account of mishaps. 

• Headings generally cost $4 to $5 per cubic yard for excava* 
tion, while the remainder of the cross-section in the same tunnel 
may cost about half as much. The average cost of a large 
number of tunnels in this country may be seen from the follow- 
ing table:* 





Cost per cubic yard. 


Cost per 
lineal foot. 




Excavation. 


Masonry. 




Material. 


Single. 






Single. 


Double. 


Single. 


Double. 


Double. 


Hard rock 

Loose rock 

Soft ground.. . . 


$5.89 
3.12 
3.62 


$5.45 

3.48 
4.64 


$12.00 

9.07 

15.00 


$8.25 
10.41 
10.50 


$69.76 

80.61 

135.31 


$142.82 
119.26 
174.42 



* Figures derived from Drinker's "Tunneling.'' 



206 RAILROAD CONSTRUCTION. § 175. 

A considerable variation from these figures may be foimd in 
individual cases, due sometimes to unusual skill (or the lack of 
it) in prosecuting the work, but the figures will generally be 
sufficiently accurate for prehminary estimates or for the com- 
parison of two proposed routes. 



CHAPTER VI. 
CULVERTS AND MINOR BRIDGES. 

176, Definition and object. Although a variable percentage 
of the rain falHng on any section of country soaks into the 
ground and does not immediately reappear, yet a very large 
percentage flows over the surface, always seeking and following 
the lowest channels. The roadbed of a railroad is constantly 
intersecting these channels, which frequently are normally dry. 
In order to prevent injury to railroad embankments by the im- 
pounding of such rainfall, it is necessary to construct waterways 
through the embankment through which such rainflow may 
freely pass. Such waterways, called culverts, are also appli- 
cable for the bridging of very small although perennial streams, 
and therefore in this work the term culvert will be apphed to 
all water-channels passing through a railroad embankment 
which are not of sufficient magnitude to require a special struc- 
tural design, such as is necessary for a large masonry arch or a 
truss bridge. 

177. Elements of- the design. A well-designed culvert must 
afford such free passage to the water that it will not ^^back up" 
over the adjoining land nor cause any injury to the embankment 
or culvert. The ability of the culvert to discharge freely all the 
water that comes to it evidently depends chiefly on the area of 
the waterway, but also on the form, length, slope, and materials 
of construction of the culvert and the nature of the approach 
and outfall. When the embankment is very low and the amount 
of water to be discharged very great, it sometimes becomes 
necessary to allow the water to discharge '^ under a head," i e., 
with the surface of the water above the top of the culvert. 
Safety then requires a much stronger construction than would 
otherwise be necessary to avoid injury to the culvert or embank- 
ment by washing. The necessity for such construction should 
be avoided if possible. 

207 



208 BAILROAD CONSTRUCTION. § 178. 



AREA OF THE WATERWAY. 

178. Elements involved. The determination of the required 
area of the waterway involves such a multiplicity of indeter- 
minate elements that any close determination of its value from 
purely theoretical considerations is a practical impossibilityo 
The principal elements involved are: 

a. Rainfall. The real test of the culvert is its capacity to 
discharge without injury the flow resulting from the extraordi- 
nary rainfalls and ^' cloud bursts" that may occur once in many 
years. Therefore, while a knowledge of the average annual 
rainfall is of very little value, a record of the maximum rainfall 
during heavy storms for a long term of years may give a relative 
idea of the maximum demand on the culvert. 

b. Area of watershed. This signifies the total area of country 
draining into the channel considered. When the drainage area 
is very small it is sometimes included within the area surveyed 
by the preliminary survey. When larger it is frequently possi- 
ble to obtain its area from other maps with a percentage of 
accuracy sufficient for the purpose. Sometimes a special survey 
for the purpose is considered justifiable. 

c. Character of soil and vegetation. This has a large in- 
fluence on the rapidity with which the rainflow from a given 
area will reach the culvert. If the soil is hard and impermeable 
and the vegetation scant, a heavy rain will run off suddenly, 
taxing the capacity of the culvert for a short time, while a 
spongy soil and dense vegetation will retard the flow, making it 
more nearly uniform and the maximum flow at any one time 
much less. 

d. Shape and slope of watershed. If the watershed is very 
long and narrow (other things being equal), the water from the 
remoter parts will require so much longer time to reach the 
culvert that the flow will be comparatively uniform, especially 
when the slope of the whole watershed is very low. When the 
slope of the remoter portions is quite steep it may result in the 
nearly simultaneous arrival of a storm- flow from all parts of the 
watershed, thus taxing the capacity of the culvert. 

e. Effect of design of culvert. The principles of h}'draulics 
show that the slope of the culvert, its length, the form of the 
cross-section, the nature of the surface, and the form of the 



§ 179. CULVERTS AND MINOR BRIDGES. 209 

approach and discharge all have a considerable influence on the 
area of cross-section required to discharge a given volume of 
water in a given time, but unfortunately the combined hy- 
draulic effect of these various details is still a very uncertain 
quantity. 

179. Methods of computation of area. There are three pos- 
sible methods of computation. 

(a) Theoretical. As shown above it is a practical impossi- 
bility to estimate correctly the combined effect of the great mul- 
tiplicity of elements which influence the final result. The nearest 
approach to it is to estimate by the use of empirical formula? 
the amount of water which will be presented at the upper end 
of the culvert in a given time and then to compute, from the 
principles of hydraulics, the rate of flow through a culvert of 
given construction, but (as shown in § 178, e) such methods are 
still very unreliable, owing to lack of experimental knowledge. 
This method has apparently greater scientific accuracy than 
other methods, but a little study will show that the elements 
of uncertainty are as great and the final result no more reliable. 
The method is most reliable for streams of uniform flow, but 
it is under these conditions that method (c) is most useful. The 
theoretical method will not therefore be considered further. 

(b) Empirical. As illustxated in § 180, some formulae make 
the area of waterway a function of the drainage area, the for- 
mula bemg affected by a coefficient the value of which is esti- 
mated between limits according to the judgment. Assuming 
that the formulae are sound, their use only narrows the limits of 
error, the final determination depending on experience and 
judgment. 

(c) From observation. This method, considered b}^ far the 
best for permanent work, consists m observing the high-water 
marks on contracted channel-openings which are on the same 
stream and as near as possible to the proposed culvert. If the 
country is new and there are no such openings, the wisest plan 
is to bridge the opening by a temporary structure in wood which 
has an ample waterway (see § 126, h, 4) and carefully observe 
all high- water marks on that opening during the 6 to 10 years 
w^hich is ordinarily the minimum life of such a structure. As 
shown later, such observations may be utilized for a close com- 
putation of the required waterway. Method (b) may be utilized 
for an approximate calculation for the required area for the tern- 



210 RAILROAD CONSTRUCTION. § ISO. 

porary structure, using a value which is intentionally excessive, 
so that a permanent structure of sufficient capacity may subse- 
quently be constructed within the temporary structure. 

1 80. Empirical formulae. Two of the best known empirical 
formulae for area of the waterway are the following: 

(a) Myer's formula: 

Area of waterway in square feet=(7Xv drainage area in acres, 
where C is a coefficient varying from 1 for flat country to 4 for 
mountainous country and rocky ground. As an illustration, if 
the drainage area is 100 acres, the waterway area should be from 
10 to 40 square feet, according to the value of the coefficient 
chosen. It should be noted that this formula does not regard 
the great variations in rainfall in various parts of the world nor 
the design of the culvert, and also that the final result depends 
largely on the choice of the coefficient. 

(b) Talbot's formula: 

Area of waterway in square feet = C X^^Cdrainage area in acres) ^. 
** For steep and rocky ground C varies from f to 1. For rolling 
agricultural country subject to floods at times of melting snow, 
and with the length of the valley three or four times its width, C 
is about i; and if the stream is longer in proportion to the area, 
decrease C, In districts not affected by accumulated snow, and 
where the length of the valley is several times the width, \ or i, 
or even less, may be used. C should be increased for steep side 
slopes, especially if the upper part of the valley has a much 
greater fall than the channel at the culvert." * As an illus- 
tration, if the drainage area is 100 acres the area of waterway 
should be 0X31.6. The area should then vary from 5 to 31 
square feet, according to the character of the country. Like 
the previous estimate, the result depends on the choice of a 
coefficient and disregards local variations in rainfall, except as 
they may be arbitrarily allowed for in choosing the coeffi- 
cient. 

181. Value of empirical formulae. The fact that these for- 
mulae, as well as many others of similar nature that have been 
suggested, depend so largely upon the choice of the coefficient 
shows that they are valuable "more as a guide to the judgment 
than as a working rule," as Prof. Talbot explicitly declares in 

* Prof. A. N. Talbot, "Selected Papers of the Civil Engineers' Club of 
the Univ. of Illinois," 



§ 182. CULVERTS AND MINOR BRIDGES. 211 

commenting on his own formula. In short, they are chiefly valu- 
able in indicating a probable maximum and minimum between 
which the true result probably lies. 

182. Results based on Observation. As already indicated in 
§ 179, observation of the stream in question gives the most 
reliable results. If the country is new and no records of the 
flow of the stream during heavy storms has been taken, even 
the life of a temporary wooden structure may not be long enough 
to include one of the unusually severe storms which must be 
allowed for, but there will usually be some high-water mark 
which will indicate how much opening will be required. The 
following quotation illustrates this: ''A tidal estuary may gen- 
erally be safely narrowed considerably from the extreme water 
lines if stone revetments are used to protect the bank from 
wash. Above the true estuary, v\^here the stream cuts through 
the marsh, we generally find nearly vertical banks, and we are 
safe if the faces of abutments are placed even with the banks. 
In level sections of the country, where the current is sluggish, 
it is usually safe to encroach somewhat on the general width 
of the stream, but in rapid streams among the hills the width 
that the stream has cut for itself through the soil should not be 
lessened, and in ravines carrying mountain torrents the open- 
ings must be left very mu^h larger than the ordinary appear- 
ance of the banks of the stream would seem to make neces- 
sary.'' * 

As an illustration of an observation of a storm-flow through 
a temporary trestle, the following is quoted : ^' Having the flood 
height and velocity, it is an easy matter to determine the vol- 
ume of water to be taken care of. I have one ten-bent pile 
trestle 135 feet long and 24 feet high over a spring branch that 
ordinarily runs about six cubic inches per second. Last sum- 
mer during one of our heavy rainstorms (four inches in less 
than three hours) I visited this place and found by float obser- 
vations the surface velocity at the highest stage to be 1.9 feet 
per second. I made a high-water mark, and after the flood- 
water receded found the width of stream to be 12 feet and an 
average depth of 2f feet. This, with a surface velocity of 1.9 
feet per second, would give approximately a discharge of 50 



* J. P. Snow, Boston & Maine Railway. From Report to Association of 
Railway Superintendents of Bridges and Buildings. 1897. 



212 RAILROAD CONSTRUCTION. § 183. 

cubic feet^ or 375 gallons, per second. Having this information 
it is easy to determine size of opening required." * 

183. Degree of accuracy required. The advantages result- 
ing from the use of standard designs for culverts (as well as 
other structures) have led to the adoption of a comparatively 
small number of designs. The practical use made of a compu- 
tation of required waterway area is to determine which one of 
several standard designs will most nearly fulfill the require- 
ments. For example, if a 24-inch iron pipe, having an area of 
3.14 square feet, is considered to be a little small, the next size 
(30-inch) would be adopted; but a 30-inch pipe has an area of 
4.92 square feet, which is 56% larger. A similar result, except 
that the percentage of difference might not be quite so marked, 
will be found by comparing the areas of consecutive standard 
designs for stone box culverts. 

The advisability of designing a culvert to withstand any 
storm-flow that may ever occur is considered doubtful. Several 
years ago a record-breaking storm in New England carried 
away a very large number of bridges, etc., hitherto supposed 
to be safe. It was not afterward considered that the design of 
those bridges was faulty, because the extra cost of constructing 
bridges capable of withstanding such a flood, added to interest 
for a long period of years, would be enormously greater than the 
cost of repairing the damages of such a storm once or twice in 
a century.* Of course the element of danger has some weight, 
but not enough to justify a great additional expenditure, for 
common prudence would prompt unusual precautions during 
or immediately after such an extraordinary storm. 

PIPE CULVERTS. 

184. Advantages. Pipe culverts, made of cast iron or earthen- 
ware, are very durable, readily constructed, moderately cheap, 
will pass a larger volume of water in proportion to the area than 
many other designs on account of the smoothness of the sur- 
face, and (when using iron pipe) may be used very close to 
the track when a low opening of large capacity is required. 
Another advantage lies in the ease with which they may be 
inserted through a somewhat larger opening that has been 

* A. J. Kelley, Kansas City Belt Railway. From Report to Association 
of Railway Superintendents of Bridges and Buildings. 1897. 



§ 185. CULVERTS AND MINOR BRIDGES. 213 

temporarily lined with wood, without disturbing the roadbed 
or track. 

185. Construction. Permanency requires that the founda- 
tion shall be firm and secure against being washed out. To 
accomplish this, the soil of the trench should be hollowed out to 
fit the lower half of the pipe, making suitable recesses for the 
bells. In very soft treacherous soil a foundation-block of con- 
crete is sometimes placed under each joint, or even throughout 
the whole length. When pipes are laid through a slightly 
larger timber culvert great care should be taken that the pipes 
are properly supported, so that there will be no settling nor 
development of unusual strains w^hen the timber finally decays 
and gives way. To prevent the washing away of material 
around the pipe the ends should be protected by a bulkhead. 
This is best constructed of masonry (see Fig. 97), although wood 
is sometimes used for cheap and minor constructions. The joints 
should be calked, especially when the culvert is hable to run 
full or when the outflow^ is impeded and the culvert is liable to 
be partly or wholly filled during freezing weather. The cost of 
a calking of clay or even hydraulic cement is insignificant com- 
pared with the value of the additional safety afforded. When 
the grade of the pipe is perfectly uniform, a very low rate of 
grade will suffice to drain a pipe culvert, but since some uneven- 
ness of grade is inevitable through uneven settlement or im- 
perfect construction, a grade of 1 in 20 should preferably be 
required, although much less is often used. The length of a 
pipe culvert is approximately determined as follows: 

Length = 2s (depth of embankment) + {width of roadbed), 

in which s is the slope ratio (horizontal to vertical) of the banks. 
In practice an even number of lengths should be used which will 
equal cr exceed the length given by this formula. 

186. Iron-pipe culverts. Simple cast-iron pipes are used in 
sizes from 12'' to 48" diameter. These are usually made in 
lengths of 12 feet with a few lengths of 6 feet, so that any required 
length may be more nearly obtained. The lightest pipes made 
are sufficiently strong for the purpose, and even those which 
would be rejected because of incapacity to withstand considerable 
internal pressure may be utilized for this work. In Fig. 97 are 
shown the standard plans used on the C. C. C. & St. L. Ry., 
which may be considered as typical plans. 




-a4 



^-^ 



Fto. 97. — Stand.'VRd Cast-iron 
Pipe Ci i.vert. C. a C. & 
St. L. Hy. (May 1893.) 



214 



§187. 



CULVERTS AND MINOR BRIDGES. 



215 



Pipes formed of cast-iron segments have been used up to 12 
feet diameter. The shell is then made comparatively thin, but 
is stiffened by ribs and flanges on the outside. The segments 
break joints and are bolted together through the flanges. The 
joints are made tight by the use of a tarred rope, together with 
neat cement. 

187. Tile-pipe culverts. The pipes used for this purpose 
vary from 12" to 24" in diameter. When a larger capacity is 
required two or more _pipes may be laid side by side, but in 
such a case another design might be preferable. It is frequently 
specified that "double-strength'' or "extra-heavy'" pipe shall 
be used, evidently with the idea that the stresses on a culvert- 
pipe are greater than on a sewer-pipe. But it has been con- 
clusively demonstrated that, no matter how deep the embank- 
ment, the pressure cannot exceed a somewhat uncertain maxi- 
mum, also that the greatest danger consists in placing the pipe 
so near the ties that shocks may be directly transferred to the 
pipe without the cushioning effect of the earth and ballast. 
When the pipes are w^ell bedded in clear earth and there is a 




UP-STHEAfVLEMD. DOWN-STREAM ETsiD. DOWN-STREAM EMD. THREE PIPE&.^ 

Fig. 98. — Standard Vitrified-pipe Culvert. Plant System. (1891.) 

sufficient depth of earth over them to avoid direct impact (at 
least three feet) the ordinary sewer-pipe will be sufficiently 
strong. ''Double-strength" pipe is frequently less perfectly 
burnod, and the supposed extra strength is not therefore ob- 



216 



BAILROAD CONSTRUCTION. 



§■ 188. 



tained. In Fig. 98 are shown the standard plans for vitrified- 
pipe culverts as used on the '^ Plant system." Tile pipe is much 
cheaper than iron pipe, but is made in much shorter lengths and 
requires much more work in laying and especially to obtain a 
uniform grade. 

BOX CULVERTS. 

1 88. Wooden box culverts. This form serves the purpose 
of a cheap temporary construction which allows the use of a 
ballasted roadbed. As in all temporary constructions, the area 
should be made considerably larger than the calculated area 
§§ 179-182), not only for safety but also in order that, if the 
smaller area is demonstrated to be sufficiently large, the per- 
manent construction (probably pipe) may be placed inside with- 
out disturbing the embankment. All designs agree in using 
heavy timbers (12''Xl2'', 10''Xl2'', or 8''Xl2'0 for the side 
walls, cross-timbers for the roof, every fifth or sixth timber 
being notched down so as to take up the thrust of the side walls, 
and planks for the flooring. Fig. 99 shows some of the standard 
designs as us^d by the C, M, & St. P. Ry. 




Fig. 



-Standard Timber Box Cut.vert. 
(Feb. 1889.) 



M. & St. p. Ry. 



189. Stone box culverts. In localities where a good quality 
of stone is cheap, stone box culverts are the cheapest form of 
permanent construction for culverts of medium capacity, but 
their use is decreasing owing to the frequent difficulty in obtain- 
ing really suitable stone within a reasonable distance of the 
culvert. The clear span of the cover-stones varies from 2 to 4 
feet. The required thickness of the cover-stones is sometimes 



§ 189. 



CULVERTS AND MINOR BRIDGES. 



217 



calculated by the theory of transverse strains on the basis of 
certain assumptions of loading — as a function of the height of 
the embankment and the unit strength of the stone used. Such 
a method is simply another illustration of a class of calculations 
which look very precise and beautiful, but which are worse than 
useless (because misleading) on account of the hopeless uncer- 




PLAN 

Fig. 100.— Standard Single Stone Culvert (3'X4'). N. & W. R.R. 

(1890.) 

tainty as to the true value of certain quantities which must be 
used in the computations In the first place the true value of 
the unit tensile strength of stone is such an uncertain and variable 



218 



KAILEOAD CONSTRUCTION. 



§189. 



quantity that calculations based on any assumed value for it are 
of small reliability. In the second place the weight of the prism 
of earth lying directly above the stone, plus an allowance for live 
load, is by no means a measure of the load on the stone nor of 
the forces that tend to fracture it. All earthwork will tend to 







r^ 




ti^ra 




IS 



PLAN 



Fig. 100a. — Standard Double Stone Culvert (3'X4'). N. &W. R. R. 

(1890.) 

form an arch above any cavity and thus relieve an uncertain 
and probably variable proportion of the pressure that might 
otherwise exist. The higher the embankment the less the pro- 



190. 



CULVERTS AND MINOR BRIDGES. 



219 



portionate loading, until at some uncertain height an increase 
in height will not increase the load on the cover-stones. The 
effect of frost is likewise large, but uncertain and not computable. 
The usual practice is therefore to make the thickness such as 
experience has shown to be safe with a good quahty of stone, 
i.e., about 10 or 12 inches for 2 feet span and up to 16 or 18 
inches for 4 feet span. The side walls should be carried down 
deep enough to prevent their being undermined by scorn* or 
heaved by frost. The use of cement mortar is also an important 
feature of first-class work, especially when there is a rapid scour- 
ing current or a liability that the culvert will run under a head. 
In Figs. 100 and 100a are shown standard plans for single and 
doublejstone box culverts as used on the Norfolkjand Western R.R. 
190. Old-rail culverts. It sometimes happens (although very 
rarely) that it is necessary to bring the grade hne mthin 3 or 4 
feet of the bottom of a stream and yet allow an area of 10 or 12 
square feet. A single large pipe of sufficient area could not be 
used in this case. The use of several smaller pipes side by side 
would be both expensive and inefficient. For similar reasons 
neither wooden nor stone box culverts could be used. In such 
cases, as well as in many others where the head-room is not so 
limited, the plan illustrated in Fig. 101 is a very satisfactory 




Fig. 101.—- Standard Old-rail Culvert. N. & W. R.R. (1895.) 

solution of the problem. The old rails, having a length of 8 or 
9 feet, are laid close together across a 6-foot opening. Some- 
times the rails are held together by long bolts passing through 



220 RAILROAD CONSTRUCTION. § 190. 

the webs of the rails. In the plan shown the rails are confined 
by low end walls on each abutment. This plan requires only 
15 inches between the base of the rail and the top of the culvert 
channel. It also gives a continuous ballasted roadbed. 

1 90a. Reinforced Concrete Culverts. The development of 
reinforced concrete as a structural material is illustrated in its 
extensive adoption for arches and also for culverts. One of the 
special types which has been adopted is that of a box culvert 
which has a concrete bottom. Since this bottom can be made 
so that it will withstand an upward transverse stress, it furnishes 
a broad foundation for the whole culvert, and thus entirely 
eliminates the necessity for extensive footing to the side walls of 
the culvert, such as are necessary in soft ground with an ordinary 
stone culvert. Another advantage is that the inside of the cul- 
vert may be made perfectly smooth and thus offer less resistance 
to the passage of water through it. As may be noticed from. 
Fig. 101a, such a culvert is provided with flaring head walls, and 
sunken end walls, so that the water may not scour underneath 
the culvert, and other features common to other types. No 
attempt will here be made to discuss the design of reinforced 
concrete, except to say that all four sides of such a box culvert 
are designed to withstand a computed bursting pressure which 
tends to crush the flat sides inward. In Fig. 101a is shown one 
illustration of the many types of culverts which have been de- 
signed of reinforced concrete. 



ARCH CULVERTS. 

191. Influence of design on flow. The variations in the design 
of arch culverts have a very marked influence on the cost and 
efficiency. To combine the least cost with the greatest effi- 
ciency, due weight should be given to the following elements : 
(a) amount of masonry, (b) the simplicity of the constructive 
work, (c) the design of the wing walls, (d) the design of the 
junction of the wing walls with the barrel and faces of the arch, 
and (e) the safety and permanency of the construction. These 
elements are more or less antagonistic to each other, and the 
defects of most designs are due to a lack of proper proportion 
in the design of these opposing interests. The simplest con- 
struction (satisfying elements h and e) is the straight barrel arch 



STANDARD ARCH CULVERT 

ePEETSRAN 

IJORFOLK & WESTERN R.R. 

(1891) 



X—t. 




MESS HAY Bi USED 

{To face page 221.) 



§ 192. 



CULVERTS AND MINOR BRIDGES. 



221 





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T-i--{--t*r- 

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222 



RAILROAD CONSTRUCTION. 



§191. 



between two parallel vertical head walls, as sketched in Fig. 
102, a. From a hydraulic standpoint the design is poor, as the 
water eddies around the corners, causing a great resistance 
which decreases the flow. Fig. 102, h, shows a much better de- 




(a) ^ (5) 

Fig. 102. — Types of Culverts, 




sign in many respects, but much depends on the details of the 
design as indicated in elements {h) and {d). As a general thing 
a good hydraulic design requires complicated and expensive 
masonry construction, i.e., elements {h) and {d) are opposed. 
Design 102, c, is sometimes inapplicable because the water is 
liable to work in behind the masonry during floods and perhaps 
cause scour. This design uses less masonry than (a) or (6). 

192. Example of arch culvert design. In Plate IV is shown 
the design for an 8-foot arch culvert according to the standard 
of the Norfolk and Western R. R. Note that the plan uses the 
flaring wing walls (Fig. 102, h) on the up-stream side (thus 
protecting the abutments from scour) and straight wing walls 
(similar to Fig. 102, c) on the down-stream end. This econo- 
mizes masonry and also simplifies the constructive work. Note 
also the simplicity of the junction of the wing walls with the 
barrel of the arch, there being no re-entrant angles below the 
springing line of the arch. The design here shown is but one 
of a set of designs for arches varying in span from 6' to 30'. 



MINOR OPENINGS. 

193. Cattle-guards, (a) Pit guards. Cattle-guards will be 
considered under the head of minor openings, since the old- 
fashioned plan of pit guards, which are even now defended and 



§193. 



CULVERTS AND MINOR BRIDGES. 



preferred by some railroad men, requires a break in the con- 
tinuity of the roadbed. A pit about three feet deep, five feet 




Fig. 103. — Cattle-quard with Wooden Slats. 



long, and as wide as the width of the roadbed, is walled up with 
stone (sometimes with wood), and the rails are supported on 
heavy timbers laid longitudinally with the rails. The break in 
the continuity of the roadbed produces a disturbance in the 
elastic wave running through the rails, the effect of which is 
noticeable at high velocities. The greatest objection, however, 
lies in the dangerous consequences of a derailment or a failure 
of the timbers owing to unobserved decay or destruction by 
fire — caused perhaps by sparks and cinders from passing loco- 
motives. The very insignificance of the structure often leads 
to careless inspection. But if a single pair of wheels gets off the 
rails and drops into the pit, a costly wreck is inevitable. 

(b) Surface cattle-guards. These are fastened on top of the 
ties ; the continuity of the roadbed is absolutely unbroken and 
thus is avoided much of the danger of a bad wreck owing to a 
possible derailment. The device consists essentially of overlay- 
ing the ties (both inside and outside the rails) with a surface on 



?24 



RAILROAD CONSTRUCTION, 



§193. 



which cattle will not walk. The multitudinous designs for such 
a surface are variously effective in this respect. An objection, 




Fig. 104.— Sheffield Cattle-guard:. 



CENTER SECTION. 




Fig. 105. — Climax Cattle-guard (tile). 

which is often urged indiscriminately against all such designs, is 
the liability that a brake-chain which may happen to be drag- 
ging may catch in the rough bars which are usod. The bars 



§ 195. CULVERTS AND MINOR BRIDGES. 225 

are sometimes "home-made/' of wood, as shown in Fig. 103. 
Steel guards may be made as shown in Fig. 104. The general 
construction is the same as for the wooden bars. The metal 
bars have far greater durability, and it is claimed that they are 
more effective in discouraging cattle from attempting to cross. 

194. Cattle-passes. Frequently when a railroad crosses a 
farm on an embankment, cutting the farm into two parts, the 
railroad company is obliged to agree to make a passageway 
through the embankment sufficient for the passage of cattle and 
perhaps even farm-wagons. If the embankment is high enough 
so that a stone arch is practicable, the initial cost is the only 
great objection to such a construction; but if an open wooden 
structure is necessary, all the objections against the old-fashioned 
cattle-guards apply with equal force here. The avoidance of a 
grade crossing which would othermse be necessary is one of the 
great compensations for the expense of the construction and 
maintenance of these structures. The construction is some- 
times made by placing two pile trestle bents about 6 to 8 feet 
apart, supporting the rails by stringers in the usual way, the 
special feature of this construction being that the embankments 
are filled in behind the trestle bents, and the thrust of the em- 
bankments is mutually taken up through the stringers, w^hich 
are notched at the ends or otherT\dse constructed so that they 
may take up such a thrust. The designs for old-rail culverts 
and arch culverts are also utilized for cattle-passes when suitable 
and convenient, as well as the designs illustrated in the folloT\dng 
section, and the reinforced concrete design of § 190a. 

195. Standard stringer and I-beam bridges. The advantages 
of standard designs apply even to the covering of short spans 
with wooden stringers or with I beams — especially since the 
methods do not require much vertical space between the rails 
and the upper side of the clear opening, a feature which is often 
of prime importance. These designs are chiefly used for cul- 
verts or cattle-passes and for crossing over highways — providing 
such a narrow opening would be tolerated. The plans all imply 
stone abutments, or at least abutments of sufficient stability to 
withstand all thrust of the embankments. Some of the designs 
are illustrated in Plate V. The preparation of these standard 
designs should be attacked by the same general methods as 
already illustrated in § 156. When computing the required 



226 RAILROAD CONSTRUCTION. § 195. 

transverse strength, due allowance should be made for lateral 
bracing, which should be amply provided for. Note particu- 
larly the methods of bracing illustrated in Plate V. The designs 
calling for iron (or steel) stringers may be classed as permanent 
constructions, which are cheap, safe, easily inspected and main- 
tained, and therefore a desirable method of construction. 




STANDARD LBRIDGES-H-FT. SPAN. 

NORFOLK AND WESTERN R.R. ^ 
(■1891.) 




5 %s 



^^-^m^—m — 



TYPES OF PLATE GIRDER BRIDGES. 

0. M. & St.P. RY. 




(ro face page 226. 



^ 



CHAPTER VH. 
BALLAST. 

196. Purpose and requirements. "The object of the ballast 
is to transfer the applied load over a large surface ; to hold the 
timber work in place horizontally; to carry off the rain-water 
from the superstructure and to prevent freezing up in winter; 
to afford means of keeping the ties truly up to the grade line; 
and to give elasticity to the roadbed/' This extremely con- 
densed statement is a description of an ideally perfect ballast. 
The value of any given kind of ballast is proportional to the 
extent to which it fulfills these requirements. The ideally 
perfect ballast is not necessarily the most economical ballast 
for all roads. Light traffic generally justifies something cheaper, 
but a very common error is to use a very cheap ballast when a 
small additional expenditure would procure a much better 
ballast, which would be much more economical in the long run. 

197. Materials. The materials most commonly employed are 
gravel and broken stone. In many sections of the country 
other materials which more or less perfectly fulfill the require- 
ments as given above, are used. The various materials includ- 
ing some of these special types have been defined by the American 
Railway Engineering and Maintenance of Way Association as 
follows: 

Definitions. 

Ballastc Selected material placed on the roadbed ior the 
purpose of holding the track in line and surface. 

Broken or crushed stone. Stone broken by artificial means 
into small fragments of specified sizes. 

Burnt clay. A clay or gumbo which has been burned into 
material for ballast. ■ 

227 



228 RAILROAD CONSTRUCTION. § 197. 

Chats. Tailings from mills in which zinc and lead ores are 
separated from the rocks in which they occur. 

Chert. An impure flint or hornstone occurring in beds. 

Cinders. The residue from the fuel used in locomotives and 
other furnaces. 

Gravel. Small worn fragments of rock, coarser than sand, 
occurring in natural deposits. 

Gumbo. A term commonly used for a peculiarly tenacious 
clay, containing no sand. 

Sand. Any hard, granular, comminuted rock material, finer 
than gravel and coarser than dust. 

Slag. The waste product, in a more or less vitrified form, of 
furnaces for reduction of ore. Usually the product of a blast- 
furnace. 

There is still another classification which may or may not be 
considered as ballast. It is perhaps hardly correct to speak 
of the natural soils as ballast, yet many miles of cheap rail- 
ways are "ballasted'' with the natural soil, which is then called 
Mud ballast. 

Broken or crushed stone. Rock ballast is generally specified 
to be that which may all be passed through a IJ inch (or 2 inch) 
ring, but which cannot pass through a |-inch mesh. It is most 
easily handled with forks. This method also has the advantage 
that when it is being rehandled the fine chips which would 
interefere with effectual drainage will be screened out. Rock 
ballast is more expensive in first cost and is also more trouble- 
some to handle, but in heavy traffic especially, the track will be 
kept in better surface and will require less work for maintenance 
after the ties have become thoroughly bedded. 

Etsmt clay. This material has been used in many sections of 
the country where broken stone or gravel are unobtainable 
except at a prohibitive cost, and where a suitable quality of 
clay is readily obtained. This clay should be of *' gumbo'* 
variety and contain no gravel. It is sometimes burnt in a 
kiln, or it is sometimes burnt by piling the clay in long heaps 
over a mass of fuel, the pile being formed in such a way that 
a temporm-y but effectual kiln is made. It is necessary that 
a clear, clean fuel shall be used and that the firing shall be 
done by a man who is experienced in maintaining such a fire 
until the burning is completed. Such bo^Hast may be burned 
very hard and it will last from four to six years. The cost of 



§ 197. BALLAST. 229 

burning varies from 30 to 60 cents per cubic yard, according 
to the circumstances. 

Chats. This is a form of ballast which is peculiar to South- 
western Missouri and Southeastern Kansas. When this mate- 
rial was first used it was obtained from the refuse piles of the 
mills which treated the zinc and lead ores mined in those regions. 
With the processes then employed the material was obtained 
in lumps as large as broken stone, and they were considered to 
be as valuable as broken stone for ballast. Improvements in 
the processes of treating the ores have resulted in making this 
by-product very much smaller grained and of less value as bal- 
last, although it is still considered a desirable form of ballast 
where it may readily be obtained. It should be noted that it 
is classed with gravel and cinders in the forms of cross-section 
shown later. 

Chert. This is a form of flint or hornstone which occurs in 
nodules of a size that is suitable for ballast, and is a very good 
type of ballast wherever it is found, but its occurrence is com- 
paratively infrequent. It is classed with cemented gravel in 
the design of cross-sections of ballast. 

Cinders. This is one of the most universal forms of ballast, 
since it is a by-product of every road which uses coal as fuel. 
The advantages consist in the fairly good drainage, the ease of 
handling and the cheapness — after the road is in operation. 
One of the greatest disadvantages is the fact that the cinders are 
readily reduced to dust, which in dry weather becomes very 
objectionable. Cinders are usually considered preferable to 
gravel in yards. 

GraveL This is one of the most common forms of good 
ballast. There are comparatively few railroads which cannot 
find, at some place along their line, a gravel pit which will 
afford a suitable supply of gravel for ballast. Sometimes it is 
unnecessary to screen it; but usually it is better to screen the 
gravel over a screen having a J-inch mesh so as to screen 
out all the dirt and the finer stones. 

Sand. Hailroads which run along the coast are frequently 
ballasted merely with the sand obtained in the immediate 
neighborhood. One great advantage lies in the almost perfect 
drainage which is obtained. 

Slag. When slag is readily obtainable it furnishes an ex- 
cellent ballast which is free from dust and perfect in drainage 



230 RAILROAD CONSTRUCTION. § 197. 

qualities. Slag is classified with crushed rock in the cross- 
sections shown below, but it should be noted that this only 
applies to the best qualities of slag, since its quality is quite 
variable. 

Mud ballast. When the natural soil is gravelly so that rain 
will drain through it quickly, it will make a fair roadbed for 
light traffic, but for heavy traffic, and for the greater part of 
the length of most roads, the natural soil is a very poor material 
for ballast ; for, no matter how suitable the soil might be along 
limited sections of the road, it would practically never happen 
that the soil would be uniformly good throughout the whole 
length of the road. Considering that a heavy rain will in one 
day spoil the results of weeks of patient '* surfacing" with mud 
ballast, it is seldom economical to use "mud" if there is a 
gravel-bed or other source of ballast anywhere on the line of 
the road. 

198. Cross-sections. The requked depth of the cross-section 
to the sub-soil depends largely on the weight of the rolling 
stock which is to pass over the track. A careful examination 
of a roadbed to determine the changes which take place under 
tho tiea and also an examination of the track and ties during 
the passage of a heavy train shows that the heavy loads which 
are now common on railroad tracks force the tie into the bal- 
last with the passage of every wheel load. The effect on the 
ballast is a greater or less amount of crushing of the ballast. 
Even the> very hardest grades of broken stone are more or less 
crushed by grinding against each other during the passage of a 
train. The softer and weaker forms of ballast are ground up 
much more quickly. One result is the formation of a fine dust 
which interferes with the proper drainage of water through the 
j^ballast. A second result is the compression of the ballast imme- 
diately under the tie into the sub-soil. In a comparatively 
short time a hole is formed under the tie which acts virtually 
like a pump. With every rise and fall of the tie under each 
wheel load, the tie actually pumps the water from the surround- 
ing ballast and sub-soil into these various holes. When the 
ballast is of such a character that the water does not drain 
through it easily, the water will settle in these holes long enough 
to seriously deteriorate the ties. When the track becoLnes so 
much out of line or level, or so loose that it needs to be tamped 
up, the process of tamping has practically the effect of deepen- 



§ 198. BALLAST. 231 

ing the amount of ballast immediately under the tie, while the 
sub-soil is forced up between the ties. A longitudinal section 
of the sub-soil of a track which has been frequently tamped 
generally has a saw-tooth appearance, and the sub-soil, instead 
of being a uniform line, has a high spot between each tie, while 
the ballast is considerably below its normal level immediately 
under the tie. 

The variation in the traffic on railroads has caused the Amei 
ican Railway Engineering and Maintenance of Way Association 
to divide railroads into three classes with respect to the stand- 
ards of construction which should be adopted for ballasting, 
as well as other details of construction. The three classes are 
as follows (quoted from the Association Manual): 

"Class *A' shall include all districts of a railway having more 
than one main track, or those districts of a railway having a 
single main track with a traffic that equals or exceeds the follow- 
ing: 

Freight-car mileage passing over districts per year per 

mile 150,000 

or, 
Passenger-car mileage per annum per mile of district. . . 10,000 

with maximum speed of passenger-trains of 50 miles per hour. 
"Class 'B' shall include all districts of a railway having a 
single main track with a traffic that equals or exceeds the 
following: 

Fieight-car mileage passing over districts per year per 

mile 60,000 

or, 
Passenger-car mileage per annum per mile of district ... 5,000 

with maximum speed of passenger-trains of 40 miles per hour. 

"Class 'C shall include all districts of a railway not meeting 
the traffic requirements of Classes 'A' or 'B.'" 

The classification was adopted on the consideration that 
quality of traffic as well as mere tonnage should determine 
the classification of a railroad. For example, it is considered 
that a road which operates a train at a speed of 50 miles an 
hour should adopt the first class or Class "A'' standards, even 
though there is but one train per day on that railroad. It 



232 



RAILROAD CONSTETTCTION. 



§198. 



likewise means that any road whose traffic makes necessary the 
construction of a regular double track should adopt the first- 
class specifications. 



-10 0- 



-13'0- 






J^ to the foot " 



I 



^ I^ 



I 



Sod«^ 






Provide drains where needed 

CRUSHED ROCK AND SLAG 



Select coarse stone 
for end of drain 




CRUSHED ROCK AND SLAG 




Provide drains where needed 



GRAVEL, CINDERS, CHATS, ETC., 



Select coarse stone 
for end of drain 



Sod 




GRAVEL, CINDERS, CHATS, EXCj, 
Fig. 106. — Cross-sections of Ballast for Class "A ' Roads. 

Tn Fig. 106 are shown a series of cross-sections which were 
recommended by that association for Class "A'' traffic. It 



§ 198. BALLAST. 233 

Bhould be noticed that in eacli case tlie cross-section of the 
roadbed from shoulder to shoulder of the roadbed is 20 feet 
plus the space between track centers for double track if any. 
The width of side ditches is merely added to that of the roadbed. 
The clear thickness of the ballast underneath the ties is made 
12 inches, but even this should be considered as the minimum 
depth and is recommended for use only on the firmest, most 
substantial and well-drained subgrades. The slope of f inch 
to the foot from the center of the track to the end of the tie, 
which is common to all the cross-sections, is designed with the 
idea of allowing a clear space of 1 inch underneath the rail. 
The ballast is then rounded off on a curve of 4 feet radius and 
finally reaches the subsoil on a slope which is 1^:1 for broken 
stone, and 3:1 for all other materials. The flat slope adopted 
for gravel, etc., which adds considerably to the required width 
of roadbed, has been so designed in order that the considerable 
mass of material at the ends of the ties shall be better able to 
hold the track in place laterally. The sod on the embank- 
ment over the shoulder of the roadbed up to within 12 inches 
of the edge of the ballast is strongly recommended on account 
of the protection it affords to the shoulder of the roadbed. 
It should be noticed that the latest decision of that associa- 
tion regarding the form of subgrade is that the subgrade should 
be made level and not crowned, as suggested and discussed in 
§63. 

In Fig. 107 are shown a series of cross-sections for various 
classes of ballast for railroads that belong to Class *'B." It 
may be noted that the thickness of the ballast under the tie 
is 9 inches for this class. The width of roadbed between the 
shoulders, recommended for Class "B" is 16 feet. As before, 
the width of the ditches is supposed to be added to this width. 
It should be noted that when using cementing gravel and chert 
the slope of 3:1 is made to begin at the bottom of the tie in- 
stead of at a point about 2 inches below the top of the tie. 
This is done in order to prevent water from accumulating 
around the end of the tie in a material which is less permeable 
than the other forms of ballast. 

In Fig. 108 are shown two cross-sections for ballast for roads 
belonging to Class '^C." On roads of this class it is assumed 
that crushed rock will not be used for ballast. The width of 



234 



RAILROAD CONSTRUCTION. 



§198. 



roadbed between shoulders is 14 feet, while the depth of ballast 
underneath the tie is 6 inches. 

It should be noticed that the above sections issued by the 
association do not include any cross-section which is recom- 
mended when no special ballast is used other than the natural 



Slope j/to the foot 




Crushed rock and slag 




Gravel, cinders, chats, etc 




Cementing gravel and chert. 
Pi«. 107.— Cross-sections op Ballast for Class " B *' Roads. 

soil. In such a case a cross-section very similar to the sec- 
tions shown for cementing gravel and chert should be used. The 
essential feature of such a section is that the soil, which is 
probably not readily permeable, should be kept away from 
the ends of the ties. Specifications for the placing of mud 
ballast, as well as other forms of ballast, have frequently speci- 
fied that the ballast should be crowned about 1 inch above the 
level of the tops of the ties in the center of the track. This 



§199. 



BALLAST. 



235 



feature of any cross-section, although proposed, was rejected 
by the association, in spite of the fact that when a tie is so 
imbedded it certainly will have a somewhat greater holding 
power in the ballast. 




Gravel, cinders, chats, etc. 




y////////////Mm 
Cementing gravel and chert. 

Fig. 108. — Cross-sections of Ballast for Class "C" Roads. 



199. Methods of laying ballast. The cheapest method of 
laying ballast on new roads is to lay ties and rails directly on 
the prepared subgrade and run a construction train over the 
track to distribute the ballast. Then the track is lifted up until 
sufficient ballast is worked under the ties and the track is prop- 
erly surfaced. This method, although cheap, is apt to injure 
the rails by causing bends and kinks, due to the passage of 
loaded construction trains when the ties are very unevenly and 
roughly supported, and the method is therefore condemned and 
prohibited in some specifications. The best method is to draw 
in carts (or on a contractor's temporary track) the ballast that | 
is required under the level of the bottom of the ties. Spread 
this ballast carefully to the required surface. Then lay the ties 
and rails, which will then have a very fair surface and uniform 
support. A construction train can then be run on the rails and 
distribute sufficient additional ballast to pack around and 
between the ties and make the required cross-section. 
^ The necessity for constructing some lines at an absolute 



236 RAILROAD CONSTRUCTION. § 200. 

minimum of cost and of opening tliem for traffic as soon as 
possible has often led to the policy of starting traffic when 
there is little or no ballast — perhaps nothing more than a mere 
tamping of the natural soil under the ties. When this is done 
ballast may subsequently be drawn where required by the train- 
load on flat cars and unloaded at a minimum of cost by means 
of a "plough/' The plough has the same width as the cars and 
is guided either by a ridge along the center of each car or by 
short posts set up at the sides of the cars. It is drawn from one 
end of the train to the other by means of a cable. The cable is 
sometimes operated by means of a small hoisting-engine car- 
ried on a car at one end of the train. Sometimes the locomo- 
tive is detached temporarily from the train and is run ahead 
with the cable attached to it. 

200. Cost- The cost of ballast in the track is quite a variable 
item for different roads, since it depends (a) on the first cost of 
the material a^ it comes to the road, (&) on the distance from 
the source of supply to the place where it is used, and (c) on 
the method of handling. The first cost of cinder or slag is 
frequently insignificant. A gravel-pit may cost nothing except 
the price of a little additional land beyond the usual limits of 
the right of way. Broken stone will usually cost SI or more 
per cubic yard. If suitable stone is obtainable on the com- 
pany's land, the cost of blasting and breaking should be some- 
what less than this. The cost of hauling will depend on the 
distance hauled, and also, to a considerable extent, on the limi- 
tations on the operation of the train due to the necessity of keep- 
ing out of the way of regular trains. There is often a needless 
waste in this way. The ^^mud train'' is considered a pariah and 
entitled to no rights whatever, regardless of the large daily cost 
of such a train and of the necessary gang of men. The cost of 
broken-stone ballast in the track is estimated at $1.25 per cubic 
yard. The cost of gravel ballast is estimated at 60 c. per cubic 
yard in the track. The cost of placing and tamping gravtl 
ballast is estimated at 20 c. to 24 c. per cubic yard, for cinders 
12 c. to 15 c. per cubic yard. The cost of loading gravel on 
cars, using a steam-shovel, is estimated at 6 c. to 10 c. per 
cubic yard.* 

• Report Roadmasters* Association. 1885. 



CHAPTER VIII. 
TIES, 
AND OTHER FORMS OF RAIL SUPPORT. 

201. Various methods of supporting rails. It is necessary 
that the rails shall be sufQciently supported and braced, so that 
the gauge shall be kept constant and that the rails shall not be 
subjected to excessive transverse stress. It is also preferable 
that the rail support shall be neither rigid (as if on solid rock) 
nor too yielding, but shall have a uniform elasticity throughout. 
These requirements are more or less fulfilled by the following 
methods. 

(a) Longitudinals. Supporting the rails throughout their 
entire length. This method is very seldom used in this country 
except occasionally on bridges and in terminals when the 
longitudinals are supported on cross-ties. In § 224 will be 
described a system of rails, used to some extent in Europe, 
having such broad bases that they are self-supporting on the 
ballast and are only connected by tie-rods to maintain the gauge. 

(b) Cast-iron "bowls" or "pots." These are castings resem- 
bling large inverted bowls or pots, having suitable chairs on 
top for holding and supporting the rails, and tied together with 
tie-rods. They will be described more fully later (§ 223). 

(c) Cross-ties of metal or wood. These will be discussed in 
the following sections. 

202. Economics of ties. The true cost of ties depends on the 
relative total cost of maintenance for long periods of time. The 
first cost of the ties delivered to the road is but one item in the 
economics of the question. Cheap ties require frequent renew- 
als, which cost for the labor of each renewal practically the 
same whether the tie is of oak or of hemlock- Cheap ties make 
a poor roadbed w^hich will require more track labor to keep even 
in tolerable condition. The roadbed will require to be disturbed 
so frequently on account of renewals that the ties never get an 
opportunity to get settled and to form a smooth roadbed for any 
length of time. Irregularity in width, thickness, or length of 
ties is especially detrimental in causing the ballast to act and 
wear unevenly. The life of ties has thus a more or less direct 
influence en the life of the rails, on the wear of rolling stock, and 
on the speed of trains. These last items are not so readily 
reducible to dollars and cents, but when it can be shown that 
the total cost, for a long period of time, of several renewals of 

237 



238 



RAILROAD CONSTRUCTION. 



§202. 



cheap ties, with all the extra track labor involved, is as great as 
or greater than that of a few renewals of durable ties, then there 
is no question as to the real economy. In the following dis- 
cussions of the merits of untreated ties (either cheap or costly), 
chemically treated ties, or metal ties, the true question is there- 
fore of the ultimate cost of maintaining any particular kind of 
ties for an indefinite period, the cost including the first cost of 
the ties, the labor of placing them and maintaining them to 
surface, and the somewhat uncertain (but not therefore non- 
existent) effect of frequent renewals on repairs of rolling stock, 
on possible speed, etc. 

WOODEN TIES. 

203. Choice of wood. This naturally depends, for any partic- 
ular section of country, on the supply of wood which is most 
readily available. The woods most commonly used, especially 
in this country, are oak and pine, oak being the most durable 
and generally the most expensive. Redwood is used very ex- 
tensively in California and proves to be extremely durable, so 
far as decay is concerned, but it is very soft and is much injured 
by "rail-cutting.'' This defect is being partly remedied by the 
use of tie-plates, as will be explained later. Cedar, chestnut, 
hemlock, and tamarack are frequently used in this country. In 
tropical countries very durable ties are frequently obtained from 
the hard woods peculiar to those countries. According to a 
bulletin of the U. S. Department of Agriculture (Forestry 
service, No. 124) the number and value of the cross-ties used 
by the steam and street railroads of the United States during 
the year 1906, was as follows: 



Kind of wood. 


Number of 
ties. 


Per cent. 


Total value. 


Aver, value 


Oaks 


45,357,874 

18,841,210 

7,248,562 

7,083.442 

6,588,975 

5,104,496 

3,969,605 

2,576,859 

2,058,198 

1,248,629 

554,738 

373,387 

1,828,067 


44.1 
18.3 
7.1 
6.9 
6.4 
5.0 
3.9 
2.5 
2.0 
1.2 
0.5 
0.3 
1.8 


$23,278,052 

9,567,745 

3,010,392 

3.310,116 

2,995,942 

1,862,135 

1,698,027 

889,561 

582,968 

536,172 

210,818 

151,052 

726,144 


$0.51 


Southern pines 

Douglas fir 


.51 

.42 


Cedar 


.47 


Chestnut 


.49 


Cypress 


.36 


Western pine 

Tamarack 


.43 
.35 


Hemlock 


.28 


Redwood 


.43 


Lodgepole pine 

White pine 


.38 
.40 


All others 


.40 






Total 


102,834,042 


100.0 


S48,819,124 


$0.47 



§ 205. TIES. . 239 

The limitations of timber supply have somewhat dimin- 
ished the use of oak and increased the use of the softer woods 
in recent years. 

204. Durability. The durability of ties depends on the cli- 
mate; the drainage of the ballast; the volume, weight, and 
speed of the traffic; the curvature, if any; the use of tie-plates; 
the time of year of cutting the timber; the age of the timber 
and the degree of its seasoning before placing in the track; the 
nature of the soil in which the timber is grown; and, chiefly, 
on the species of wood employed. The variability in these 
items will account for the discrepancies in the reports on the life 
of various woods used for ties. 

White oak is credited with a life of 5 to 12 years, depending 
principally on the traffic. It is both hard and durable, the 
hardness enabling it to withstand the cutting tendency of the 
rail-flanges, and the durability enabling it to resist decay. P^ne 
and redwood resist decay very well, but are so soft that they are 
badly cut by the rail-flanges and do not hold the spikes very 
well, necessitating frequent respiking. Since the spikes must 
be driven within certain very limited areas on the face of each 
tie, it does not require many spike-holes to ''spike-kill" the 
tie. On sharp curves, especially w^ith heavy traffic, the wheel- 
flange pressure produces a side pressure on the rail tending to 
overturn it, which tendency is resisted by the spike, aided some- 
times by rail-braces. Whenever the pressure becomes too great 
the spike wdll yield somewhat and will be shghtly withdrawn. 
The resistance is then somewhat less and the spike is soon so 
loose that it must be redriven in a new hole. If this occurs 
very often, the tie may need to be replaced long before any decay, 
has set in. When the traffic is very light, the wood very dura- 
ble, and the cHmate favorable, ties have been known to last 
25 years. 

205. Dimensions. The usual dimensions for the best roads 
(standard gauge) are 8' to 9' long, 6'' to 7'' thick, and 8'' to 
10'' wide on top and bottom (if they are hewed) or 8'' to 9'' 
wide if they are sawed. For cheap roads and light traffic the 
length is shortened sometimes to 7' and the cross-section also 
reduced. On the other hand a very few roads use ties 9' 6" long. 

Two objections are urged against sawed ties: £rst, that the 
grain is torn by the saw, leaving a woolly surface which induces 
decay; and secondly, that, since timber is not perfectly straight- 



240 RAILROAD CONSTRUCTION. § 205. 

grained, some of the fibers are cut obliquely, exposing their ends, 
which are thus liable to decay. The use of a ^^ planer-saw" ob- 
viates the first difficulty. Chemical treatment of ties obviates 
both of these difficulties. Sawed ties are more convenient to 
handle, are a necessity on bridges and trestles, and it is even 
claimed, although against commonly received opinion, that 
actual trial has demonstrated that they are more durable than 
hewed ties. 

206. Spacing. The spacing is usually 14 to 16 ties to a 30- 
foot rail. This number is sometimes reduced to 12 and even 
10, and on the other hand occasionally increased to 18 or 20 by 
employing narrower ties. There is no economy in reducing the 
number of ties very much, since for any required stiffness of 
track it is more economical to increase the number of supports 
than to increase the weight of the rail. The decreasing cost of 
rails and the increasing cost of ties have materially changed the 
relation between number of ties and weight of rail to produce a 
given stiffness at minimum cost, but many roads have found it 
economical to employ a large number of ties rather than increase 
the weight of the rail. On the other hand there is a practical 
limit to the number that may be employed, on account of the 
necessary space between the ties that is required for proper 
tamping. This width is ordinarily about twice the width of the 
tie. At this rate, with light ties- 6'' wide and with 12" clear 
space, there would be 20 ties per 30-foot rail, or 3520 per mile. 
The smaller ties can generally be bought much cheaper (propor- 
tionately) than the larger sizes, and hence the economy. 

Track instructions to foremen generally require that the 
spacing of ties shall not be uniform along the length of any 
rail. Since the joint is generally the weakest part of the rail 
structure, the joint requires more support than the center of the 
rail. Therefore the ties are placed with but 8'' or 10'' clear 
space between them at the joints, this applying to 3 or 4 ties at 
each joint; the remaining ties, required for each rail length, are 
equally spaced along the remaining distance. 

207. Specifications. The specifications for ties are apt to 
include the items of size, kind of wood, and method of construc- 
tion, besides other minor directions about time of cutting, sea- 
soning, delivery, quality of timber, etc. 

(a) Size. The particular size or sizes required will be some- 
what as indicated in § 205. 



§ 208. TIES. 241 

(b) Kind of wood. When the kind or kinds of wood are 
specified^ the most suitable kinds that are available in that 
section of country are usually required. 

(c) Method of construction. It is generally specified that the 
ties shall be hewed on two sides; that the two faces thus made 
shall be parallel planes and that the bark shall be removed. It 
is sometimes required that the ends shall be sawed off square; 
that the timber shall be cut in the winter (when the sap is down) ; 
and that the ties shall be seasoned for six months These last 
specifications are not required or lived up to as much as their 
importance deserves. It is sometimes required that the ties shall 
be delivered on the right of way, neatly piled in rows, the alter- 
nate rows at right angles, piled if possible on ground not lower 
than the rails and at least seven feet away from them, the lower 
row of ties resting on two ties which are themselves supported 
so as to be clear of the ground. 

(d) Quality of timber. The usual specifications for sound 
timber are required, except that they are not so rigid as for a 
better class of timber work The ties must be sound, reason- 
ably straight-grained, and not very crooked — one test being that 
a line joining the center of one end with the center of the middle 
shall not pass outside of the other end. Splits or shakes, espe- 
cially if severe, should causa rejection. 

Specifications sometimes require that the ties shall be cut 

from single trees, making 
what is known as ^^pole 
p^^J ties'^ and definitely con- 
demning- those which are 

POLE TIE. SLAB TIE. QUARTER TIE. *^ 

„ ^^^ __ ,^ cut or split from larger 

FiQ. 109. — Methods of cutting Ties. , , : . , ., f i 

trunks, givmg two ^ slab 

ties'' or four " quarter ties" for each cross-section, as is illustrated 
in Fig. 109. Even if pole ties are better, their exclusive use 
means the rapid destruction of forests of young trees. 

2o8. Regulations for laying and renewing ties. The regula- 
tions issued by railroad companies to their track foremen will 
generally include the following, in addition to directions regard- 
ing dimensions, spacing, and specifications given in §§ 204-207. 
When hewn ties of somewhat variable size are used, as is fre- 
quently the case, the largest and best are to be selected for use 
as joint ties. If the upper surface of a tie is found to be warped 
(contrary to the usual specifications) so that one or both rails do 




242 RAILROAD CONSTRUCTION. § 208. 

not get a full bearing across the whole width of the tie, it must 
be adzed to a true surface along its whole length and not mereJs' 
notched for a rail-seat. When respiking is necessary and spikes 
have been pulled out, the holes should be immediately plugged 
with "wooden spikes," which are supplied to the foreman for 
that express purpose, so as to fill up the holes and prevent the 
decay which would otherwise take place when the hole becomes 
filled with rain-water. Ties should always be laid at right angles 
to the rails and never obliquely Minute regulations to prevent 
premature rejection and renewal of ties are frequently made. It 
is generally required that the requisitions for renewals shall be 
made by the actual count of the individual ties to be renewed 
instead of by any wholesale estimates. It is unwise to have ties 
of widely variable size, hardness, or durability adjacent to each 
other in the track, for the uniform elasticity, so necessary for 
smooth riding, will be unobtainable under those circumstances. 

209. Cost of ties. When railroads can obtain ties cut by 
farmers from woodlands in the immediate neighborhood, the 
price will frequently be as low as 20 c for the smaller sizes, 
running up to 50 c for the larger sizes and better qualities, espe- 
cially when the timber is not very plentiful Sometimes if a 
railroad cannot procure suitable ties from its immediate neigh- 
borhood, it will find that adjacent railroads control all adjacent 
sources of supply for their own use and that ties* can only be 
procured from a considerable distance, with a considerable added 
cost for transportation . First-class oak ties cost about 75 to 80 c. 
and frequently much more Hemlock ties can generally be 
obtained for 35 c. or less. 

PRESERVATIVE PROCESSES FOR WOODEN TIES. 

210. General principle. Wood has a fibrous cellular struc- 
ture, the cells being filled with sap or air. The woody fiber is 
but little subject to decay unless the sap undergoes fermentation. 
Preservative processes generally aim at lemovmg as much of the 
water and sap as possible and filling up the pores of the wood 
with an antiseptic compound The most common methods (ex- 
cept one) all agree in this general process and only differ in the 
method employed to get rid of the sap and in the antiseptic 
chemical with which the fibers are filled One valuable. feature 
of these processes lies in the fact that the softer cheaper woods 



^ 



§ 212. TIES, 243 

(such as hemlock and pine) are more readily treated than are the 
harder woods and yet will produce practically as good a tie as a 
treated hard-wood tie and a very much better tie than an un- 
treated hard-wood tie. The various processes will be briefly 
described, taking up first the process which is fundamentally 
different from the others, viz., vulcanizing. 

211. Vulcanizing. The process consists in heating the timber 
to a temperature of 300° to 500° F. in a cylinder, the air being 
under a pressure of 100 to 175 lbs. per square inch. By this 
process the albumen in the sap is coagulated, the water evapo* 
rated, and the pores are partially closed by the coagulation of 
the albumen. It is claimed that the heat sterilizes the wood and 
produces chemical changes in the wood which give it an antisep- 
tic character. It was once very extensively used on the ele- 
vated lines of New York City, but the process has now been 
abandoned as unsatisfactory. 

212. Creosoting. — This process consists in impregnating the 
wood with wood-creosote or with dead oil of coal-tar. Wood-- 
creosote is one of the products of the destructive distillation of 
wood — usually long-leaf pine. Dead oil of coal-tar is a prod- 
uct of the distillation of coal-tar at a temperature between 480° 
and 760° F. It would require about 35 to 50 pounds of creo- 
sote to completely fill the pores of a cubic foot of w^ood But 
it would be impossible to force such an amount into the wood, 
nor is it necessary or desirable. About 10 pounds per cubic 
foot, or about 35 pounds per tie, is all that is necessary. For 
piling placed in salt water about 18 to 20 pounds per cubic foot 
is used, and the timber is then perfectly protected against the 
ravages of the teredo navalis. To do the work, long cylinders, 
which may be opened at the ends, are necessary. Usually the/ 
timbers are run in and out on iron carriages running on rails 
fastened to braces on the inside of the cylinder. When the load 
has been run in, the ends of the cylinder are fastened on. The 
water and air in the pores of the wood are first drawn out by 
subjecting the wood alternately to steam- pressure and to the 
action of a vacuum-pump. This is .continued for several hours. 
Then, after one of the vacuum periods, the cylinder is filled 
with creosote oil at a temperature of about 170° F. The pumps 
are kept at work until the pressure is about 80 to 100 pounds 
per square inch, and is maintained at this pressure from one to 
two hours according to the size of the timber. The oil is then 



244 RAILROAD CONSTRUCTION. § 213o 

withdrawn, the cylinders opened, the train pulled out and an- 
other load made up in 40 to 60 minutes. The average time re- 
quired for treating a load is about 18 or 20 hours, the absorption 
about 10 or 11 pounds of oil per cubic foot, and the cost (1894) 
from $12.50 to $14.50 per thousand feet B. M, 

213. Burnettizing (chloride-of-zinc process). This process is 
very similar to the creosoting process except that the chemical is 
chloride of zinc, and that the chemical is not heated before use. 
The preliminary treatment of the wood to alternate vacuum and 
pressure is not continued for quite so long a period as in the 
creosoting process. Care must be taken, in using this process, 
that the ties are of as uniform quality as possible, for seasoned 
ties will absorb much more zinc chloride than unseasoned (in the 
same time), and the product will lack uniformity unless the sea- 
soning is uniform. The A., T. & S. Fe R. R. has works of its 
own at which ties are treated by this process at a cost of about 
25 c. per tie. The Southern Pacific R. R. also has works for 
burnettizing ties at a cost of 9.5 to 12 c per tie The ziuL- 
chloride solution used in these works contains only 1.7% of zinc 
chloride instead of over 3% as used in the Santa Fe works, which 
perhaps accounts partially for the great difference in cost per tie. 
One great objection to burn etti zed ties is the fact that the chem- 
ical is somewhat easily washed out, when the wood again be- 
comes subject to decay Another objection, which is more 
forcible with respect to timber subject to great stresses, as in 
trestles, than to ties, is the fact that when the solution of zinc 
chloride is made strong (over 3%) the timber is made very brittle 
and its strength is reduced. The reduction in strength has been 
shown by tests to amount to J to ^^ of the ultimate strength, 
and that the elastic limit has been reduced by about ^. 

214. Kyanizing (bichloride-of-mercuiy or corrosive-sublimate 
process). This is a process of ''steeping/' It requires a much 
longer time than the previously described processes^ but docs not 
require such an expensive plant. Wooden tanks of sufficient 
size for the timber are all that is necessary. The corrosive subli- 
mate is first made into a concentrated solution of one part of 
chemical to six parts of hot water When used in the tanks this 
solution is weakened to 1 part in 100 or 150. The w^ood will 
absorb about 5 to G.5 pounds of the bichloride per 100 cubic 
feet, or about one pound for each 4 to 6 ties. The timber is 
allowed to soak in the tanks for several days, the general rule 



§ 215. TIES. 245 

being about one day for each inch of least thickness and one day 
over — which means seven days for six-inch ties, or thirteen (to 
fifteen) days for 12'' timber (least dimension). The process is 
somewhat objectionable on account of the chemical being such a 
virulent poison, workmen sometimes being sickened by the fumes 
arising from the tanks. On the Baden railway (Germany) 
kyanized ties last 20 to 30 years. On this railway the wood is 
always air-dried for two weeks after impregnation and before 
being used, w^hich is thought to have an important effect on its 
durability. The solubility of the chemical and the liability of 
the chemical w^ashing out and leaving the wood unprotected is 
an element of Aveakness in the method. 

215. Wellhouse (or zinc- tannin) process. The last two 
methods described (as well as some others employing similar 
chemicals) are open to the objection that since the wood is im- 
pregnated with an aqueous solution, it is liable to be washed out 
very rapidly if the wood is placed under water, and will even 
disappear, although more slowly, under the action of moisture 
and rain. Several processes have been proposed or patented to 
prevent this. Many of them belong to one class, of which the 
Wellhouse process is a sample. By these processes the timber 
is successively subjected to the action of two chemicals, each 
individually soluble in water, and hence readily impregnating 
the timber, but the chemicals when brought in contact form in- 
soluble compounds w^hich cannot be washed out of the wood- 
cells. By the Wellhouse process, the wood is first im^pregnated 
with a solution of chloride of zinc and glue, and is then subjected 
to a bath of tannin under pressure. The glue and tannin com- 
bine to form an insoluble leathery compound in the cells, which 
will prevent the zinc chloride from being washed out. It is 
being used by the A., T. &. S. Fe R. R., their works being 
located at Las Vegas, New Mexico, and also by the Union 
Pacific R. R. at their works at Laramie, Wyo. In 1897 Mr. J. 
M. Meade, a resident engineer on the A., T. & S. Fe, exhibited 
to the Roadmasters Association of America a piece of a tie treated 
by this process which had been taken from the tracks after 
nearly 13 years' service. The tie w^as selected at random, was 
taken out for the sole purpose of having a specimen, and w^as 
still in sound condition and capable of serving many years longer. 
The cost of the treatment was then quoted as 13 c. per tie. 



246 RAILROAD CONSTRUCTION, § 216. 

It was claimed that the treatment trebled the life of the tie 
besides adding to its spike-holding power. 

In spite of this apparently favorable showing, the process 
was abandoned on the A. T. & S. F. R. R. in 1898 on the 
ground that the results did not justify the added expense. 

2 1 6. Cost of Treating. The cost of treating ties by the vari- 
ous methods has been estimated as follows * — assuming that 
the plant was of sufficient capacity to do the work economi- 
cally: creosoting, 25 c. per tie; vulcanizing, 25 c. per tie; 
burnettizing (chloride of zinc), 8.25 c. per tie; kyanizing (steep- 
ing in corrosive sublimate), 14.6 c. per tie; Wellhouse process 
(chloride of zinc and tannin), 11.25 c. per tie. These estimates 
are only for the net cost at the works and do not include the 
cost of hauling the ties to and from the works, which may mean 
5 to 10 c. per tie. Some of these processes have been installed 
on cars which are transported over the road and operated where 
most convenient. An estimate made in 1907 by Prof. Gellert 
AUeman on the cost of treating ties, each assumed to have a 
volume of 3 cubic feet, the cost ''not including royalty on pa- 
tents, profit, interest, or depreciation, all of which vary widely 
at the various plants, '' is as follows: 

Zinc chloride . . o 16 cents 

" " and creosote 27 " 

Creosote, 10 pounds to the cubic foot , . , 55 " 

The very grert increase in these prices, especially for creosot- 
ing, is due to the enormous increase in late years in the con- 
sumption and in the price of creosote. 

217. Economics of treated ties. The fact that treated ties are 
not universally adopted is due to the argument that the added 
life of the tie is not worth the extra cost. If ties can be bought 
for 25 c, and cost 25 c. for treatment, and the treatment only 
doubles their life, there is apparently but little gained except 
the work of placing the extra tie in the track, which is more 
or less, offset by the interest on 25 c. for the life of the untreated 
tie, and the larger initial outlay makes a stronger impression on 
the mind than the computed ultimate economy. But when 
(utilizing some statistics from the Pittsburg, Ft. Wayne & 

* Bull. No. 9, V. S. Dept. of Agric, Div. of Forestry. App. No. 1, by 
Henry Flad. 



§ 217. TIES. 247 

Chicago Hailroad) it is found tliat white oak ties laid in rock 
ballast had a life of 10.17 years, and that hemlock ties treated 
with the zinc-tannin process and laid in the same kind of ballast 
lasted 10.71 years, then the economy is far more apparent. 
Unfortunately no figures were given for the cost of these ties 
nor for the cost of the treatment; but if we assume that the 
white oak ties cost 75 c. and the hemlock ties 35 c. plus 20 c. 
for treatment, there is not only a saving of 20 c. on each tie, 
but also the advantage of the slightly longer life of the treated 
tie. In the above case the total life of the two kinds of ties 
is so nearly the same that we may make an approximation of 
their relative w^orth by merely comparing the initial cost; but 
usually it is necessary to compare the value of two ties one 
of which may cost more than the other, but will last considerably 
longer. The mathematical comparison of the real value of 
two ties under such conditions may be developed as follows: 
The real cost of a tie, or any other similar item of constructive 
work, is measured by the cost of perpetually maintaining that 
item in proper condition in the structure. It will be here 
assumed that the annual cost of the trackwork, which is assign- 
able to the tie, is the same for all kinds of ties, although the 
difference probably lies in Javor of the more expensive and 
most durable ties. By assuming this expense as constant, the 
remaining expense may be considered as that due to the cost 
of the new ties whenever necessary, plus the cost of placing 
them in the track. We also may combine these two items 
in one, and consider that the cost of placing a tie in the track, 
which we will assume at the constant value of 20 c. per tie, 
regardless of the kind of tie, is merely an item of 20 c. in the 
total cost of the tie. We will assume that T^ is the present 
cost of a tie, the cost including the preservative treatment if 
any, and the cost of placing in the track. The tie is assumed 
to last n years. At the end of n years another tie is placed 
in the track, and, for lack of more precise knowledge, we will 
assume that this cost T2 equals T^, The ^'present worth'' 
of T2 is the sum which, placed at compound interest, would 
equal T2 at the end of n years, and is expressed by the quantity 
T2 



,- .^, in which r equals the rate of interest. Similarly at 

the end of 2n years we must expend a sum T^ to put in the third 
tie, and the present worth of the cost of that third tie is ex- 



248 RAILROAD CONSTRUCTION. §217. 

T 
pressed by the fraction . _J \2 n ^^ ^^^^ similarly express 

the present worths of the cost of ties for that particular spot 
for an indefinite period. The sum of all these present worths 
is given by the sum of a converging series and equals (assuming 

that all the Ts are equal) . . ^ -. But instead of laying 

aside a sum of money which will maintain a tie in that par- 
ticular place in perpetuity, we may compute the annual sum 
which must be paid at the end of each year, which would be 
the equivalent. We will call that annual payment J., and 
then the present worths of all these items are as follows: 

For the first payment rnTT' 



For the second payment n _i_-».\2» 



A 
(l+r)2 

For the third payment ■ .3 / 

For the nth payment 7-— — r-. 

^ '^ (l+r)n 

After the next tie is put in place we have the present worths 
of the annual payments on the second tie, of which the first 
one would be 

For the (n + 1) payment (i+r)in+i y 

Similarly after x ties hava been put in place the last pay- 

ment for the x tie would have a present worth -— . The 

sum of all these present worths is represented by the sum of 

a converging series and equals the very simple expression — • 

But since the sum of the present worths of these annual pay- 
ments must equal the sum of the present worths of the payments 
made at intervals of n years, we may place these two summa- 
tions equal to each other, and say that 

. rXrx(H-r)^ 

il== ^ —* 

(1+rr-l 



§217. TIES. 249 

Values of A for various costs of a tie T on the basis that r 
equals 5% have been computed and placed in Table XXXIV. 
To illustrate the use of this table, assume that we are comparing 
the relative values of two ties, both untreated, one of them 
a white oak tie which will cost, say 75 c, and will last twelve 
years, the other a yellow pine tie which will cost, say 35 c, 
and will last six years. Assuming a charge for each case of 
20 c. for placing the tie in the track, we have as the annual 
charge against the white oak tie, which costs 95 c. in the track, 
10.72 c. The pine tie, costing 55 c. in the track and lasting 
Bix years, will be charged with an annual cost of 10.48 c, which 
shows that the costs are practically equal. It is probably 
true that the track work for maintaining the white oak would 
be less than that for the pine tie, but since the initial cost of 
the pine tie is less than that of the oak tie, it would probably 
be preferred in this case, especially if money was difficult to 
obtain. It may be interesting to note that if a comparison is 
made from a similar table which is computed on the basis of 
compounding the money at 4% instead of 5%, the annual 
charges would be 10.13 and 10.49 c. for the oak and pine ties 
respectively, thus showing that when money is "easier'' the 
higher priced tie has the greater advantage. 

Example 2. Considering" again the comparison previously 
made of a white oak untreated tie which was assumed to cost 
75 c, and a hemlock treated tie, which cost 35 c. for the tie 
and 20 c. for the treatment, the total costs of these ties laid 
in the track would therefore be 95 c. and 75 c. respectively. 
These ties had practically the same life (10.17 and 10.71 years), 
but in order to use the table, we will call it ten years for each 
tie. The annual charge against the oak tie would therefore 
be 12.30 c, while that against the hemlock tie would be 9.72 c. 
This gives an advantage in the use of the treated tie of 2.58 c. 
per year, which capitalized at 5% would have a capitalized 
value of 51.6 c. 

The Atchison, Topeka and Santa Fe R. H. has compiled a 
record of treated pine ties removed in 1897, '98, '99, and 1900, 
showing that the average life of the ties removed had been about 
11 years. On the Chicago, Rock Island and Pacific R. R., the 
average life of a very large number of treated hemlocK and 
tamarack ties was found to be 10.57 years. Of one lot of ^"1,850 
ties, 12% still remained in the track after 15 years' exposure. 

It has been demonstrated that much depends on the niinoir 



250 RAILROAD CONSTRUCTION. § 218. 

details of the process — whatever it may be. As an illustra- 
tion, an examination of a batch of ties, treated by the zinc- 
creosote process, showed 84% in service after 13 years' ex- 
posure ; another batch, treated by another contractor by the 
same process (nominally), showed 50% worthless after a service 
of six years. 

METAL TIES. 

2 1 8. Extent of use. In 1894 * there were nearly 35000 miles 
of ''metal track" in various parts of the world. Of this total, 
there were 3645 miles of ''longitudinals" (see § 224), found ex- 
clusively in Europe, nearly all of it being in Germany. There 
were over 12000 miles of "bowls and plates" (see § 223), found 
almost entirely in British India and in the Argentine Republic. 
The remainder, over 18000 miles, was laid with metal cross-ties 
of various designs. There were over 8000 miles of metal cross- 
ties in Germany alone, about 1500 miles in the rest of Europe, 
over 6000 miles in British India, nearly 1000 miles in the rest 
of Asia, and about 1500 miles more in various other parts of the 
world. Several railroads in this country have tried various de- 
signs of these ties, but their use has never passed the experi- 
mental stage. These 35000 miles represent about 9% of the 
total railroad mileage of the world — nearly 400000 miles. They 
represent about 17.6% of the total railroad mileage, exclusive of 
the United States and Canada, where they are not used at all, 
except experimentally, t In the four years from 1890 to 1894 the 
use of metal track increased from less than 25000 mile3 to nearly 
35000 miles. This increase was practically equal to the total in- 
crease in railroad mileage during that time, exclusive of the 
increase in the United States and Canada. This indicates a 
large growth in the percentage of metal track to total mileage, 
and therefore an increased appreciation of the advantages to be 
derived from their use. 

219. Durability. The durability of metal tiec; is still far 
from being a settled question, due largely to the fa.ct that the 
best form for such ties is not yet determined, and that a large 
part, of the apparent failures in metal ties hr.vo been evidently 
due to defective design. Those in favor of then estimate the 
life as from 30 to 50 years. The opponents place it at not more 

* Bulletin No. 9, U. S. I>ept. of Agriculture, Div. of Forestry. 
t See § 220 for a later development. 



§ 220. TIES. 251 

than 20 years, or perhaps as long as the best of wooden ties. 
Unhke the wooden tie, however, which deteriorates as much 
with time as with usage, the hfe of a metal tie is more largely a 
function of the traffic. The life of a well-designed metal tie has 
been estimated at 150000 to 200000 trains; for 20 trains per 
day, or say 6000 per year, this would mean from 25 to 33 years. 
20 trains per day on a single track is a much larger number than 
will be found on the majority of railroads. Metal ties are found 
to be subject to rust, especially when in damp localities, such as 
tunnels; but on the other hand it is in such confined localities, 
where renewals are troublesome, that it is especially desirable to 
employ the best and longest-lived ties. Paint, tar, etc., have 
been tried as a protection against rust, but the efficacy of such 
protection is as yet uncertain, the conditions preventing any re- 
newal of the protection — such as may be done by repainting a 
bridge, for example. Failures in metal cross-ties have been 
largely due to cracks which begin at a corner of one of the square 
holes which are generally punched through the tie, the holes 
being made for the bolts by which the rails are fastened to the 
tie. The holes are generally punched because it is cheaper. 
Reaming the holes after punching is thought to be a safeguard 
against this frequent cause of failure. Another method is to 
round the corners of the square punch with a radius of about 
J". If a crack has already started, the spread of the crack may 
be prevented by drilling a small hole at the end of it. 

220. Form and dimensions of metal cross- ties. Since stabiHty 
in the ballast is an essential quality for a tie, this must be accom- 
plished either by turning down the end of the tie or by having 
some form of lug extending downward from one or more points 
of the tie. The ties are sometimes depressed in the center (see 
Plate VI, N. Y. C. & H. R. R. R. tie) to allow for a thick cover- 
ing of ballast on top in order to increase its stability in the 
ballast. This form requires that the ties should be sufficiently 
well tamped to prevent a tendency to bend out straight, thus 
widening the gauge. Many designs of ties are objectionable 
because they cannot be placed in the track without disturbing 
adjacent ties. The failure of many metal cross-ties, otherwise 
of good design, may be ascribed to too light weight. Those 
weighing much less than 100 pounds have proved too light. 
From 100 to 130 pounds weight is being used satisfactorily on 
German railroads. The general outside dimensions are about 



252 RAILROAD CONSTRUCTION. § 220. 

the same as for wooden ties, except as to thickness. The metal 
is generally from V' to I" thick. They are, of course, only made 
of wrought iron or steel, cast iron being used only for '^ bowls" or 
" plates'' (see § 223). The details of construction for some 
of the most commonly used ties may be seen by a study of 
Plate VI. 

The Carnegie tie is perhaps the only tie whose use on steam 
railroads in this country has passed the experimental stage. 
The Bessemer and Lake Erie R. R. in 1910 had 188 miles of 
track laid with these ties, and other roads are making extensive 
experiments. One practical difficulty, which is not of course 
insuperable, arises from the common practice of using the rails 
as parts of an electrical circuit for a block-signal system, which 
requires that the rails shall be insulated from each other. This 
requires that these metal ties shall be insulated from the rails. 
A method of insulation which is altogether satisfactory and 
inexpensive is yet to be determined. It is claimed that, on 
account of the better connection between the rail and the tie, 
there is less wear and more uniform wear to the rail. It is 
also claimed that there is greater lateral rigidity in the rails 
and ties (considered as a structure) and that this decreases the 
trackwork necessary to maintain alinement. These ties weigh 
19.7 pounds per linear foot, or about 167 pounds for an 8 foot 
6 inch tie. Even at the lowest possible price per pound the 
cost of the tie and its fastenings must be two or three times 
that of the best oak tie with spikes and even tie plates. It 
has been impossible to estimate the probable life of these ties. 
Until a reasonably close estimate of the life of steel ties can 
be determined, no proper comparison can be made of their 
economy relative to that of wooden ties. A study of Table 
XXXIV will show that a tie which costs, say three times as 
much as a cheap tie, must last more than three times as long 
in order that the annual charge against the tie shall be as low 
as that of the cheaper tie. For example, let us assume that 
the cost of a metal tie, laid in the track, is $2.55 and that it 
will last 20 years. From Table XXXIV we may find that the 
annual charge against $2.55 at 5% for 20 years =(2X8.02) -f 
4.41 = 20.45 c. Compared with a tie costing 65 c, plus 20 c. 
for track laying, we find that the cheaper tie will only cost 
19.63 c. per year even if it only lasts 5 years. Of course the 
claimed advantage of better track and less cost for track main- 
tenance, using steel ties, will tend to offset, so far as it is true. 




LIVESEY BOWL.(18G4) 



Plate VI.— Some Forms of Metal Ties. 
pp. 252 and 253.) 



§ 223. TIES. 253 

the disadvantage of the extra cost of the metal tie. Even 
if the extra work per tie amounts to only one-half hour for 
one man in a year, the cost of it, say 6 c., will utterly change 
the relative economics of the two ties. 

221. Fastenings. The de\ices for fastening the rails to the 
ties should be such that the gauge may be T^ddened if desired on 
curves, also that the gauge can be made true regardless of slight 
inaccuracies in the manufacture of the ties, and also that shims 
may be placed under the rail if necessary during cold weather 
when the tie is frozen into the ballast and cannot be easily 
disturbed. Some methods of fastening require that the base of 
the rail be placed against a lug which is riveted to the tie or 
which forms a part of it. This has the advantage of reducing 
the number of pieces, but is apt to have one or more of the 
disadvantages named above. Metal keys or wooden wedges are 
sometimes used, but the majority of designs employ some form 
of bolted clamp. The form adopted for the experimental ties 
used by the N. Y. C. & H. R. R. R. (see Plate VI) is especially 
ingenious in the method used to vary the gauge or allow for 
inaccuracies of manufacture. Plate VI shows some of the 
methods of fastening adopted on the principal types of ties. 

222. Cost. The cost of metal cross-ties in Germany averages 
about 1.6 c. per pound or about $1.60 for a 100-lb. tie. The ties 
manufactured for the N. Y. C. & H. R. R. R. in 1892 weighed 
about 100 lbs. and cost $2.50 per tie, but if they had been made 
in larger quantities and with the present price of steel the cost 
would possibly have been much lower. The item of freight 
from the place of manufacture to the place where used is no 
inconsiderable item of cost with some roads. Metal cross-ties 
have been used by some street railroads in this country. Those 
used on the Terre Haute Street Railway weigh 60 pounds and 
cost about 66 c. for the tie, or 74 c. per tie with the fastenings. 



223. Bowls or plates. As mentioned before, over 12000 miles 
of railway, chiefly in British India and in the Argentine Repub- 
lic, are laid with this form of track. It consists essentially of 
large cast-iron inverted ^' bowls" laid at intervals under each 
rail and opposite each other, the opposite bowls being tied 
together with tie-rods. A suitable chair is riveted or bolted on 
to the top of each bowl so as to properl}^ hold the rail. Being 



254 RAILROAD CONSTRUCTION. § 224. 

made of cast iron, they are not so subject to corrosion as steel 
or wrought iron. They have the advantage that when old and 
worn out their scrap value is from 60% to 80% of their initial 
cost, while the scrap value of a steel or wrought-iron tie is prac- 
tically nothing. Failure generally occurs from breakage, the 
failures from this cause in India being about 0.4% per annum. 
They weigh about 250 lbs. apiece and are therefore quite expen- 
sive in first cost and transportation charges. There are miles 
of them in India which have already lasted 25 years and are 
still in a serviceable condition. Some illustrations of this form 
of tie are shown in Plate VI. 

224. Longitudinals.* This form, the use of which is con- 
fined almost exclusively to Germany, is being gradually replaced 
on many lines by metal cross-ties. The system generally con- 
sists of a compound rail of several parts, the upper bearing rail 
being very light and supported throughout its length by other 
rails, which are suitably tied together with tie-rods so as to 
maintain the proper gauge, and which have a sufficiently broad 
base to be properly supported in the ballast. One great objec- 
tion to this method of construction is the 
difficulty of obtaining proper drainage espe- 
cially on grades, the drainage having a ten- 
dency to follow along the lines of the rails. 
■^zzzzzzzzzzz^ '^^^ construction is much more complicated 
on sharp curves and at frogs and switches. 
Another fundamentally different form of 
longitudinal is the Haarman compound '^ self-bearing" rail, 
having a base 12" wide and a height of 8", the alternate sections 
breaking joints so as to form a practically continuous rail. 

Some of the other forms of longitudinals are illustrated in 
Plate VI. 

For a very complete discussion of the subject of metal ties, 
see the "Report on the Substitution of Metal for Wood in 
Railroad Ties'' by E. E. Russell Tratman, it being Bulletin 
No. 4, Forestry Division of the U. S. Dept. of Agriculture. 

* Although the discussion of longitudinals might be considered to be 
long more properly to the subject of Rails, yet the essential idea of all de- 
signs must necessarily be the support of a rail-head on which the rolling 
Btock may run, and therefore this form, unused in this country, will be 
briefly described here. 



§ 224a. METAL TIES. 255 

224a. Reinforced Concrete Ties. — The wide application of 
reinforced concrete to various structural purposes, combined 
with its freedom from decay, has led to its attempted adoption 
for ties. At present (1908) their use is w^hoUy in the experi- 
mental stage. In the annual Proceedings of the American 
Railway Engineering and Maintenance of Way Association for 
1907 is a report on over a dozen different designs, the most of 
which were shewn to be incapable of enduring traffic except on 
sidings. The ties are particularly subject to fracture if struck 
by a derailed car. A similar progress report, made in 1911, 
again indicated that a practicable concrete tie for general use 
has not yet been invented. 

One of the most successful of these ties is the "Buhrer," 
which consists of one-fourth part of a thirty-foot scrap rail, 
which is inverted so that the base forms the seat of the running 
rails. This rail is imbedded in a mass of concrete whose form 
is somewhat like that of a huge ^'pole''' tie. Several thousands 
of these are in use on various roads, but many of them have 
already required renewal and none of them have yet had time 
to show a service which would make them more economical 
than wooden ties. 



CHAPTER IX. 

RAILS. 

225. Early forms. The first rails ever laid were wooden 
stringers which were used on very short tram-roads around coal- 
mines. As the necessity for a more durable rail increased, 
owing chiefly to the invention of the locomotive as a motive 
power, there were invented successively the cast-iron ** fish- 
belly" rail and various forms of wrought-iron strap rails which 
finally developed into the T rail used in this country and the 
double-headed rail, supported by chairs, used so extensively in 
England. The cast-iron rails were cast in lengths of about 3 
feet and were supported in iron chairs which were sometimes 
set upon stone piers. A great deal of the first railroad track 
of this country was laid with longitudinal stringers of v/ood 
placed upon cross-ties, the inner edge of the stringers being 
protected by wrought- iron straps. The "bridge" rails were 
first rolled in this country in 1844. The ''pear" section was 
an approach to the present form, but was very defective on 
account of the difficulty of designing a good form of joint. The 
''Stevens" section was designed in 1830 by Col. Robert L. 
Stevens, Chief Engineer of the Camden and Amboy Railroad; 
although quite defective in itf: proportions, according to the 
present knowledge of the requirements, it is essentially the pres- 
ent form. In 1836, Charles Vignoles invented essentially the 
same form in England; this form is therefore known throughout 
England and Europe as the Vignoles rail. 

226. Present standard forms. The larger part of modern 
railroad track is laid with rails v/hich are either "T" rails or 
the double-headed or " bull-headed " railc which are carried in 
chairs. The double-headed rail v/as designed with r. symmetri- 
cal form with the idea that after one head had been worn out 
by traffic the rail could be reversed, and that its life would be 
practically doubled. Experience has shown that the wear of the 

256 



I 226. 



RAILS. 



257 



rail in the chairs is very great; so much so that when one head 
has been worn out by traffic the whole rail is generally useless. 



y/P7Z\ 

BALT. & OHIO R. R. 
QUINCYR.R. 1843. 



1826. 



BULL-HEAD." 




'FISH-BELLY"— CAST IRON. 



CAST IRON. 




reynolds— 1767. 
Fig. 111. — Early Forms of Rails. 

If the rail is turned over, the worn places, caused by the chairs, 
make a rough track and the rail appears to be more brittle and 
subject to fracture, possibly due to the crystallization that may 
have occurred during the previous usage and to the reversal of 
stresses in the fibers. Whatever the explanation, experience has 
demonstrated the ]aci. The ''bull-headed'' 
rail has the lower head only large enough to 
properly hold the wooden keys with which 
the rail is secured to the chairs (see Fig. 112) 
and furnish the necessary strength. The use 
of these rails requires the use of two cast- 
iron chairs for each tie. It is claimed that 
such track is better for heavy and fast ^traffic, but it is more 




Fig. 112. — Bull- 
headed Rail and 
Chair. 



258 



RAILROAD CONSTRUCTION. 



§ 226. 



expensive to build and maintain. It is the standard form of 
track in England and some parts of Europe. 

Until a few years ago there was a very great multiplicity 
in the designs of ^'T" rails as used in this country, nearly every 
prominent railroad having its own special design, which perhaps 
differed from that of some other road by only a very minute and 
insignificant detail, but vviiich nevertheless would require a 
complete new set of rolls for rolling. This certainly must have 
had a very appreciable effect on the cost of rails. In 1893, the 
iVmerican Societ}^ of Civil Engineers, after a very exhaustive 
investigation of the subject, extending over several years, hav- 
ing obtained the opinions of the best experts of the country, 
adopted a series of sections which have been very extensively 
adopted by the railroads of this country. Instead of having 
the rail sections for various weights to be geometrically similar 
figures, certain dimensions are made constant, regardless of the 
weight. It was decided that the metal should be distributed 
through the section in the proportions of — head 42%, web 21%, 
and flange 37%. The top of the head should have a radius of 




Fig. 113. — Am. Soc. C. E. Standard Rail Section. 



12"; the top corner radius of head should be yV'; the lower 
corner radius of head should be ^^" ] the corners of the flanges, 
y\" radius; side radius of web, 12''; top and bottom radii of 
web corners, \"] and angles with the horizontal of the under side 



§227. 



RAILS. 



259 



of the head and the top of the flange, 13°. The sides of the head 
are vertical. 

The height of the rail (D) and the width of the base (C) are 
always made equal to each other. 





Weight per Yard. 




40 


45 


50 


55 


60 


65 


70 


75 


80 


85 


90 


95 


ico 


A 


IF 


2" 


2r 


2¥' 


2r 


2ir 


2tV' 


2ir 


2V 


2xV' 


2V 


2ir 


21" 


B 


M 


II 


t\ 


M 


ii 


i 


II 


11 


11 


r% 


T% 


T% 


X«5 


C &D 


3i 


3H 


Si 


4x^5 


4Jr 


4/^ 


4f 


4ii 


5 


5h 


51 


5x1 


5i 


E 


1 


M 


H 


M 


H 


M 


if 


11 


i 


II 


if 


if 


M 


F 


HI 


1§^ 


2^^ 


2\l 


2il 


2f 


2M 


2|| 


2f 


2f 


2f| 


211 


3/* 


G 


Uz 


IxV 


li 


Ui 


1^ 


Ih 


lii 


HI 


u 


IM 


HI 


m 


111 



The chief features of disagreement among railroad men relate 
to the radius of the upper corner of the head and the slope of the 
side of the head. The radius (A") adopted for the upper corner 
(constant for all weights) is a little more than is advocated by 
those in favor of "sharp corners" who often use a radius of \'\ 
On the other hand it is much less than is advocated by those 
who consider that it should be nearly equal 
to (or even greater than) the larger radius 
imiversally adopted for the corner of the 
wheel-flange. The discussion turns on the 
relative rapidity of rail wear and the wear 
of the wheel-flanges as affected by the rela- 
tion of the form of the wheel-tread to that 
of the rail. It is argued that sharp rail 
corners wear the wheel-flanges so as to 
produce sharp flanges, which are liable to 

Fig. 114. — Relation cause derailment at switches and also to 
OF Rail to Wheel- . x t_ j. xi. x* £ • j • 

TREAD. reqmre that the tires of engme-drivers 

must be more frequently turned down to their true form. On 

the other hand it is generally believed that rail wear is much less 

rapid while the area of contact between the rail and wheel-flange 

is small, and that when the rail has worn down, as it invariably 

does, to nearly the same form as the wheel-flange, the rail wears 

away very quickly. 

227. Weight for various kinds of traffic. The heaviest rails 

in regular use weigh 100 lbs. per yard, and even these are only 

used on some of the heaviest traffic sections of such roads as the 




260 RAILROAD CONSTRUCTION. § 227, 

N. Y. Central, the Pennsylvania, the N. Y., N. H, & H., and 
a few others. Probabl}^ the larger part of the mileage of the 
country is laid with 60- to 75-lb. rails — considering the fact that 
^'the larger part of the mileage'' consists of comparatively light- 
traffic roads and may exclude all the heavy trunk lines. Very 
light-traffic roads are sometimes laid with 56-lb. rails. Roads 
with fairly heavy traffic generally use 75- to 85-lb. rails, espe- 
cially when grades are heavy and there is much and sharp curva- 
ture. The tendency on all roads is toward an increase in the 
weight, rendered necessary on account of the increase in the 
weight and capacity of rolling stock, and due also to the fact that 
the price of rails has been so reduced that it is both better and 
cheaper to obtain a more solid and durable track by increasing 
the weight of the rail rather than by attempting to support a 
weak rail by an excessive number of ties or by excessive track 
labor in tamping. It should be remembered that in buying rails 
the mere weight is, in one sense, of no importance. The im- 
portant thing to consider is the strength and the stiffness. If 
we assume that all weights of rails have similar cross-sections 
(which is nearly although not exactly true), then, since for beams 
of similar cross-sections the strength varies as the cube of the 
homologous dimensions and the stiffness as the fourth power, 
while the area (and therefore the weight per unit of length) 
only varies as the square, it follows that the stiffness varies as 
the square of the \\ eight, and the strength as the | power of the 
weight. Since for ordinary variations of weight the price per 
ton is the same, adding (say) 10% to the weight (and cost) adds 
^1% to the stiffness and over 15% to the strength. As another 
illustration, using an 80-lb. rail instead of a 75-lb. rail adds only 
6|% to the cost, but adds about 14% to the stiffness and nearly 
11% to the strength. This shows why heavier rails are more 
economical and are being adopted even when they are not abso- 
lutely needed on account of heavier roUing stock. The stiffness, 
strength, and consequent durability are increased in a much 
greater ratio than the cost. 

228. Effect of stiffness on traction. A very important but 
generally unconsidered feature of a stiff rail is its effect on trac- 
tive force. An extreme illustration of this principle is seen 
when a vehicle is drawn over a soft sandy road. The constant 
compression of the sand in front of the wheel has virtually the 
same effect on traction as drawing the wheel up a grade whose 



§ 229. RAILS. 261 

steepness depends on the radius of the wheel and the depth of 
the rut. On the other hand, if a wheel, made of perfectly 
elastic material, is rolled over a surface which, while supported 
with absolute rigidity, is also perfectly elastic, there would be a 
forward component, caused by the expanding of the compressed 
metal just behind the center of contact, which would just bal- 
ance the backward component. If the rail was supported 
throughout its length by an absolutely rigid support, the high 
elasticity of the wheel-tires and rails would reduce this foi*m of 
resistance to an insignificant quantity, but the ballast and even 
the ties are comparatively inelastic. When a weak rail yields, 
the ballast is more or less compressed or displaced, and even 
though the elasticity of the rail brings it back to nearly its 
former place, the work done in compressing an inelastic material 
is wholly lost. The effect of this on the fuel account is certainly 
very considerable and yet is frequently entirely overlooked. It 
is practically impossible to compute the saving in tractive power, 
and therefore in cost of fuel, resulting from a given increase in 
the weight and stiffness of the rail, since the yielding of the rail 
is so dependent on the spacing of the ties, the tamping, etc. But 
it is not difficult to perceive in a general way that such an econ- 
omy is possible and that it' should not be neglected in considering 
the value of stiffness in rails. - 

229. Length of rails. The recommended standard minimum 
length of rails is 33 feet. In recent years many roads have been 
trying 45-foot and even 60-foot rails. The argument in favor of 
longer rails is chiefly that of the reduction in track-joints, which 
are costly to construct and to maintain and are a fruitful source 
of accidents. Mr. Morrison of the Lehigh Valley R. R.* declares 
that, as a result of extensive experience with 45-foot rails on 
that road, he finds that they are much less expensive to handle, 
and that, being so long, they can be laid around sharp curves 
without being curved in a machine, as is necessary with the 
shorter rails. The great objection to longer rails Hes in the 
difficulty in allowing for the expansion, which will require, in 
the coldest weather, an opening at the joint of nearly f for a 
60-foot rail. The Pennsylvania R. R. and the Norfolk and 
Western R. R. each have a considerable mileage laid with 60-foot 
rails. 

* Report, Roadmasters Association, 1895. 



262 RAILROAD CONSTRUCTION. § 230. 

230. Expansion of rails. Steel expands at the rate of .0000065 
of its length per degree Fahrenheit. The extreme range of tem- 
perature to which any rail will be subjected will be about 160°, 
or say from -20° F. to +140° F. With the above coefficient 
and a rail length of 60 feet the expansion would be 0.0624 foot, 
or about f inch. But it is doubtful whether there would ever 
be such a range of motion even if there were such a range of 
temperature. Mr. A. Torrey, chief engineer of the Mich. Cent. 
H. R., experimented with a section over 500 feet long, which, 
although not a single rail, was made "continuous'^ by rigid 
splicing, and he found that there was no appreciable additional 
contraction of the rail at any temperature below +20° F. The 
reason is not clear, but the fact is undeniable. 

The heavy girder rails, used by the street railroads of the 
country, are bonded together with perfectly tight rigid joints 
which do not permit expansion. If the rails are laid at a tem- 
perature of 60° F. and the temperature sinks to 0°, the rails 
have a tendency to contract .00039 of their length. If this 
tendency is resisted by the friction of the pavement in which the 
rails are buried, it only results in a tension amounting to .00039 
of the modulus of elasticity, or say 10920 pounds per square 
inch, assuming 28 000000 as the modulus of elasticity. This 
stress is not dangerous and may be permitted. If the tempera- 
ture rises to 120° F., a tendency to expansion and buckling will 
take place, which will be resisted as before by the pavement, 
and a compression of 10920 pounds per square inch will be in- 
duced, which will likewise be harmless. The range of tempera- 
ture of rails which are buried in pavement is much less than 
when they are entirely above the ground and will probably 
never reach the above extremes. Rails supported on ties which 
are only held in place by ballast must be allowed to expand and 
contract almost freely, as the ballast cannot be depended on to 
resist the distortion induced by any considerable range of tem- 
perature, especially on curves. 

231. Rules for allowing for temperature. Track regulations 
generally require that the track foremen shall use iron (not 
wooden) shims for placing between the ends of the rails while 
splicing them. The thickness of these shims should vary with 
the temperature. Some roads use such approximate rules as the 
following : " The proper thickness for coldest weather is y\ of an 
inch; during spring and fall use J of an inch, and in the very 



§ 232. 



RAILS. 



263 



hottest weather re of an inch should be allowed." This is on 
the basis of a 30-foot rail. When a more accurate adjustment 
than this is desired, it may be done by assuming some very high 
temperature (100° to 125° F.) as a maximum, when the joints 
should be tight; then compute in tabular form the spacing for 
each temperature, varying by 25°, allowing 0".0643 (very 
nearly re'O for each 25° change. Such a tabular form would. 
be about as follows (rail length 33 feet): 



Temperature . . 


Over 100° 


100°-75° 


75°-50° 


50°-25° 25°-0° 


Below 0° 


Rail opening . . . 


Close 


A" 


¥' 


^" 


r 


A"" 



One practical difficulty in the way of great refinement in this 
work is the determination of the real temperature of the rail 
when it is laid. A rail lying in the hot sun has a very much 
higher temperature than the air. The temperature of the rail 
cannot be obtained even by exposing a thermometer directly to 
the sun, although such a result might be the best that is easily 
obtainable. On a cloudy or rainy day the rail has practically 
the same temperature as the air; therefore on such days there 
need be no such trouble. 

232. Chemical composition.- About 98 to 99.5% of the com- 
position of steel rails is iron, but the value of the rail, as a rail, 
is almost wholly dependent upon the large number of other 
chemical elements which are, or may be, present in very small 
amounts. The manager of a steel-rail miU once declared that 
their aim was to produce rails having in them — 

Carbon 0.32 to 0.40% 

Silicon 0.04 to 0.06% 

Phosphorus 0.09 to 0. 105% 

Manganese 1 .00 to 1 .50% 

The analysis of 32 specimens of rails on the Chic, Mil. & St. 
Paul R. R. showed variations as follows: 



Carbon 0.211 to 0.52% 

Silicon. , 0.013 to 0.256%, 

Phosphorus 0.055 to 0. 181% 

Manganese. , . 35 to 1 . 63% 



264 



RAILROAD CONSTRUCTION. 



§ 232. 



These quantities have the same general relative proportions 
as the rail-mill standard given above, the differences lying 
mainly in the broadening of the limits. Increasing the per- 
centage of carbon by even a few hundredths of one per cent 
makes the rail harder, but likewise more brittle. If a track is 
well ballasted and not subject to heaving by frost, a harder and 
more brittle rail may be used without excessive danger of break- 
age, and such a rail will wear much longer than a softer tougher 
rail, although the softer tougher rail may be the better rail for 
a road having a less perfect roadbed. 

A small but objectionable percentage of sulphur is some- 
times found in rails, and very delicate analysis will often show 
the presence, in very minute quantities, of several other chem- 
ical elements. The use of a very small quantity of nickel or 
aluminum has often been suggested as a means of producing 
a more durable rail. The added cost and the uncertainty of 
the amount of advantage to be gained has hitherto prevented 
the practical use or manufacture of such rails. 

233. Proposed standard specifications for steel rails. The 
following specifications for steel rails are those proposed by a 
committee of the American Railway Engineering Association in 
March, 1910: 

PROCESS OF MANUFACTURE. 

1. The entire process of manufacture shall be in accordance 
with the best current state of the art. 

(a) Ingots shall be kept in a vertical position until ready to 
be rolled, oruntil the metal in the interior has had time to solidify. 
(6) Bled ingots shall not be used. 

CHEMICAL COMPOSITION. 

2. The chemical composition of the steel from which the rails 
are rolled shall be within the following limits: 





Bessemer. 


Open-hearth. 




80 lbs. and 
under. 


85 to 100 lbs. 
inclusive. 


80 lbs. and 
under. 


85 to 100 lbs. 
inclusive. 


Carbon 


0.40 to 0.50 
0.80 to 1.10 
0.10 to 0.20 

0.10 
0.075 


0.45 to 0.55 
0.85 to 1.15 
0.10 to 0.20 

0.10 
0:075 


0.53 to 0.66 
0.75 to 1.00 
0.10 to 0.20 

0.04 
0.06 


0.63 to 0.76 


Manganese 


0.75 to 1.00 


Silicon 


0.10 to 0.20 


Phosphorus not to ex- 
ceed 


0.04 


Sulphur not to exceed . 


0.06 



§ 233. RAILS. 265 

3. When lower phosphorus can be secured in Bessemer or 
open-hearth steel, the carbon shall be increased at the rate of 
0.035 for each 0.01 reduction in phosphorus. 

The percentages of carbon, manganese, and sihcon in an entire 
order of rails shall average as high as the mean percentages be- 
tween the upper and lower Hmits. 

SHEARING. 

4. There shall be sheared from the end of the bloom formed 
from the top of the ingot, sufficient discard to insure sound rails. 
All metal from the top of the ingot, whether cut from the bloom 
or the rail, is the top discard. 

SHRINKAGE. 

5. The number and passes and speed of train shall be so regu- 
lated that, on leaving the rolls at the final pass, the temperature 
of the rails will not exceed that which requires a shrinkage allow- 
ance at the hot saws, for a 33-ft. rail of 100 lbs. section of 6J ins., 
and I in. less for each 10 lbs. decrease of section, these allow- 
ances to be decreased at the rate of xJ^ iii- for each second, 
of time elapsed between the rail leaving the finishing rolls and 
being sawed. The bars shalT not be held for the purpose of 
reducing their temperature, nor shall any artificial means of 
cooling them be used between the leading and finishing passes, 
nor after they leave the finishing pass. 

SECTION. 

6. The section of rail shall conform as accurately as possible 
to the templet furnished by the railroad company. A variation 
in height of -^ in. less or ^ in. greater than the specified height, 
and Y^ in. in width of flange, will be permitted; but no variations 
shall be allowed in the dimensions affecting the fit of sphce bars. 

WEIGHT. 

7. The weight of the rail shall be maintained as nearly as pos- 
sible, after complying with the preceding paragraph, to that 
specified in the contract. 

A variation of one-half of one per cent from the calculated 



266 KAILROAD CONSTRUCTION. § 233. 

r 

weight of section, as applied to an entire order, will be 
allowed. 
Rails will be accepted and paid for according to actual weight. 

LENGTH. 

8. The standard length of rail shall be 33 ft. Ten per 
cent of the entire order will be accepted in shorter lengths varying 
as follows: 30 ft., 28 ft., and 26 ft. A variation of | in. from the 
specified length will be allowed. 

All No. 1 rails less than 33 ft. shall be painted green on both 
ends. 

STRAIGHTENING. 

9. Care shall be taken in hot-straightening rails, and it shall 
result in their being left in such condition that they will not 
vary throughout their entire length more than four (4) ins. from 
a straight line in any direction when delivered to the cold- 
straightening presses. Those which vary beyond that amount, 
or have short kinks, shall be classed as second quality rails and 
be so marked. The distance between supports of rails in the 
straightening press shall not be less than forty-two (42) ins.; 
supports to have flat surfaces and out of wind. Rails shall be 
straight in line and surface and smooth on head when finished, 
final straightening being done while cold. They shall be sawed 
square at ends, variations to be not more than ^ in., and prior 
to shipment shall have the burr caused by the saw cutting 
removed and the ends made clean. 

DRILLING. 

10. Circular holes for joint bolts shall be drilled in accordance 
with specifications of the purchaser. They shall in every respect 
conform accurately to drawing and dimensions furnished and 
shall be free from burrs. 

BRANDING. 

11. The name of the maker, the weight of the rail, and the 
month and year of manufacture shall be rolled in raised letters 
and figures on the side of the web. The number of the heat and 
a letter indicating the portion of the ingot from which the rail 
was made shall be plainly stamped on the web of each rail, where 
it will not be covered by the splice bars. Rails to be lettered 



§233. RAILS. 267 

consecutively A, B, C, etc., the rail from the top of the ingot 
being A. In case of a top discard of twenty or more per cent 
the letter A will be omitted. Open-hearth rails to be branded 

DROP TESTS. 

12. Drop tests shaU be made on pieces of rail rolled from the 

top of the ingot, not less than fom- (4) ft. and not more than six 
(6) ft. long, from each heat of steel. These test pieces shall be 
cut from the rail bar next to either end of the top rail, as selected 
by the inspector. 

The temperature of the test piece shall be between forty (40) 
and one hundred (100) degrees Fahrenheit. 

The test pieces shall be placed head upward on soHd supports, 
five (5) ins. top radius, three (3) ft. between centers, and sub- 
jected to impact tests, the tup faUing free from the following 
heights : 

60- and 70-lb. rail 16 ft. 

80-, 85-, and 90-lb. rail 18 ft. 

100-lb. rail 20 ft. 

The test pieces which do not break imder the first drop shall 
be nicked and tested to destruction. 

DEFLECTION. 

13. It is proposed to prescribe, under this head, the require- 
ments in regard to deflection, fixing maximum and minimum 
limits, as soon as proper deflection Hmits have been decided on. 

(a) Two pieces shall be tested from each heat of steel. If 
either of these test pieces breaks, a third piece shall be tested. 
If two of the test pieces break without showing physical defect, 
all rails of the heat will be rejected absolutely. If two of the 
test pieces do not break, all rails of the heat will be accepted 
as No. 1 or No. 2 classification, according as the deflection is 
less or more, respectively, than the prescribed Hmit.* 

(6) If, however, any test piece broken under test ^^A" shows 
physical defect, the top rail from each ingot of that heat shall be 
rejected. 

(c) Additional tests shall then be made of test pieces selected 
by the inspector from the top end of any second rails of the same 

* This clause to be added when the deflection limits are specified. 



267a KAILROAD CONSTRUCTION. § 233. 

heat. If two of out three of these second test pieces break, the 
remainder of the rails of the heat will also be rejected. If two out 
of three of these second test pieces do not break, the remainder 
of the rails of the heat will be accepted, provided they conform 
to the other requirements of these specifications, as No. 1 or 
No. 2 classification, according as the deflection is less or more, 
respectively, than the prescribed limit.* 

(d) If any test piece, test ^ ^A," does not break, but when nicked 
and tested to destruction shows interior defect, the top rails from 
each ingot of that heat shall be rejected. 

DROP-TESTING MACHINE. 

14. The drop-testing machine shall be the standard of the 
American Railway Engineering and Maintenance of Way Asso- 
ciation, and have a tup of 2000 lbs. weight, the striking face 
of which shall have a radius of five (5) ins. 

The anvil block shall be adequately supported and shall weigh 
20 000 lbs. 

The supports shall be a part of or firmly secured to the anvil. 

NO. 1 RAILS. 

15. No. 1 "rails shall be free from injurious defects and flaws 
of all kinds. 

NO. 2 RAILS. 

16. Rails which, by reason of surface imperfections, are not 
accepted as No. 1 rails, will be classed as No. 2 rails, but rails 
which in the judgment of the inspector contain physical defects 
which impair their strength, shall be rejected. 

No. 2 rails to the extent of five (5) per cent of the whole order 
will be received. All rails accepted as No. 2 rails must have the 
ends painted white, and shall have two prick punch marks on 
the side of the web near the heat number near the end of the 
rail, so placed as not to be covered by the sphce bars. 

Rails improperly drilled or straightened, or from which the 
burrs have not been properly removed, shall be rejected, but 
may be accepted after being properly finished. 

All classes of rails must be kept separate from each other and 
shipped in separate cars. 

All rails must be loaded in the presence of the inspector. 



* This clause to be added when the deflection limits are specified. 



§ 234. RAILS, 2676 



INSPECTION. 

17. (a) Inspectors representing the purchaser shall have free 
entry to the works of the manufacturer at all times while the 
contract is being executed, and shall have all reasonable facilities 
afforded them by the manufacturer to satisfy them that the rails 
have been made in accordance with the terms of the specifica- 
tions- 

(b) For Bessemer steel the manufacturer shall, before the rails 
are shipped, furnish the inspector daily with carbon determina- 
tions for each heat, and two complete chemical analyses every 
twenty-four hours representing the average of the other elements 
contained in the steel, for each day and night turn. These analy- 
ses shall be made on driUings taken from small test ingots. 
The drillings for analyses shall be taken from the ladle test 
ingot at a distance of J in. beneath the surface. 

For open-hearth steel, the makers shall furnish the inspectors 
with the complete chemical analysis for each melt. 

(c) On request of the inspector, the manufacturer shall furnish 
a portion of the test ingot for check analysis. 

(d) All tests and inspections shall be made at the place of 
manufacture, prior to shipment, and shall be so conducted as not 
to unnecessarily interfere with-the operation of the mill. 

(e) Rails to be accepted must meet all of the requirements 
of the specifications. 

234. Rail wear on tangents. When the wheel loads on a rail 
are abnormally heavy, and particularly when the rail has but 
little carbon and is unusually soft, the concentrated pressure 
on the rail is frequently greater than the 
elastic limit, and the metal ^'flows'' so that 
the head, although greatly abraded, ^dll 
spread somewhat outside of its original lines, 
as shown in Fig. 115. The rail wear that 
occurs on tangents is almost exclusively 
on top. Statistics show that the rate of 
rail wear on tangents decreases as the rails 
are more worn. Tests of a large number of 
rails on tangents have shown a rail wear averaging nearly one 
pound per yard per 10 000 000 tons of traffic. There is about 
33 pounds of metal in one yard of the head of an 80-lb. rail. As 
an extreme value this may be worn down one-half, thus giving 





268 RAILROAD CONSTRUCTION. § 234. 

a tonnage of 165 000 000 tons for the life of the rail. Other 
estimates bring the tonnage down to 125 000 000 tons. Since 
the locomotive is considered to be responsible for one-half (and 
possibly more) of the damage done to the rail, it is found that 
the rate of wear on roads with shorter trains is more rapid in 
proportion to the tonnage, and it is therefore thought that the 
life of a rail should be expressed in terms of the number of trains. 
This has been estimated at 300 000 to 500 000 trains. 

235. Rail wear on curves. On curves the maximum rail wear 
occurs on the inner side of the head of the outer rail, giving a 
worn form somewhat as shown in Fig. 116. The dotted line 
shows the nature and progress of the rail wear 
on the inner rail of a curve. Since the press- 
ure on the outer rail is somewhat lateral 
rather than vertical, the "flow" does not 
take place to the same extent, if at all, on 
the outside, and whatever flow would take 
place on the inside is immediately worn off 
by the wheel-flange. Unlike the wear on 
tangents, the wear on curves is at a greater 
rate as the rail becomes more worn. 

The inside rail on curves wears chiefly on top, the same as 
on a tangent, except that the wear is much greater owing^to the 
longitudinal slipping of the wheels on the rail, and the lateral 
slipping that must occur when a rigid four-wheeled truck is 
guided around a curve. The outside rail is subjected to a 
greater or less proportion of the longitudinal slipping, likewise 
to the lateral slipping^ and, worst of all, to the grinding action 
of the flange of the wheel, which grinds off the side of the head. 
The results of some very elaborate tests^ made by Mr. A M. 
Wellington, on the Atlantic and Great Western R. R., on the 
wear of rails, seem to show that the rail wear on curves may be 
expressed by the formula: "Total wear of rails on a. d degree 
curve in pounds per yard per 10 000 000 tons duty = l+0.03c?^.'' 
"It is not pretended that this formula is strictly correct even 
in theory, but several theoretical considerations indicate that 
it may be nearly so." According to this formula the average 
rail wear on a 6° curve will be about twice the rail wear on a tan- 
gent. While this is approximately true^ the various causes 
modifying the rate of rail wear (length of trains, age and quality 
of rails, etc.) will result in numerous and large variations from 



§ 236. RAILS. 269 

the above formula, which should only be taken as indicating an 
approximate law. 

236. Cost of rails. In 1 873 the cost of steel rails was about 
$120 per ton, and the cost of iron rails about $70 per ton 
Although the steel rails w^re at once recognized as superior to 
iron rails on account of more uniform wear, they were an expen- 
sive luxury. The manufacture of steel rails by the Bessemer 
process created a revolution in prices, and they steadily dropped 
in price until, several years ago, steel rails were manufactured 
and sold for $22 per ton. For several years past the price has 
been very uniform at $28 per ton at the mill. At such prices 
there is no longer any demand for iron rails, since the cost of 
manufacturing them is substantially the same as that of steel 
rails, while their durability is unquestionably inferior to that of 
steel rails. Rail quotations are generally on the basis of "long 
tons" of 2240 pounds. 

The freight charge for transporting rails from the mill to the 
place where used is usually so large that it adds a very appreciable 
amount to the cost per ton. As an approximation, the freight 
may be estimated as 0.6 c. per ton-mile, or $3.00 per ton for 
haxil of 500 miles. 



CHAPTER X. 

RAIL-FASTENINGS. 

RAIL-JOINTS 

237. Theoretical requirements for a perfect joint. A perfect 
rail-joint is one that has the same strength and stiffness — no 
more and no less — as the rails which it joins, and which will 
not interfere with the regular and uniform spacing of ties* It 
should also be reasonably cheap both in first cost and in cost of 
maintenance. Since the action of heavy loads on an elastic rail 
is to cause a wave of translation in front of each wheel, any 
change in the stiffness or elasticity of the rail structure will 
cause more or less of a shock, which must be taken up and 
resisted by the joint. The greater the change in stiffness the 
greater the shock, and the greater the destructive action of the 
shock. The perfect rail-joint must keep both rail-ends truly in 
line both laterally and vertically, so that the flange or tread of 
the wheel need not jump or change its direction of motion sud- 
denly in passing from one rail to the other. A consideration of 
all the above requirements will show that only a perfect wielding 
of rail-ends would produce a joint of uniform strength and stiff- 
ness which would give a uniform elastic wave ahead of each 
wheel. As welding is impracticable for ordinary railroad work 
(see § 230), some other contrivance is necessary w^hich will 
approach this ideal as closely as may be. 

238. Efficiency of the ordinary angle-bar. Throughout the 
middle portion of a rail the rail acts as a continuous girder. If 
we consider for simplicity that the ties are unyielding, the deflec- 
tion of such a continuous girder between the ties will be but 
one-fourth of the deflection that would be found if the rail were 
cut half-w^ay between the ties and an equal concentrated load 
were divided equally between the two unconnected ends. The 
maximum stress for the continuous girder w^ould be but one-half 
of that in the cantilevers. Joining these ends with rail-joints 
will give the ordinary ^^ suspended" joint. In order to main- 

270 



§ 239. RAIL-FASTENINGS. 271 

tain uniform strength and stiffness the angle-bars must supply 
the deficiency. These theoretical relations are modified to an 
unknown extent by the unknown and variable yielding of the 
ties From some experiments made by the Association of 
Engineers of Maintenance of Way of the P. R. R.* the following 
deductions were made: 

1. The capacity of a ^'suspended" joint is greater than that 
of a "supported" joint — whether supported on one or three 
ties. (See §240) 

2. That (^dth the particular patterns tested) the angle-bars 
alone can carry only 53 to 56% of a concentrated load placed 
on a joint. 

3. That the capacity of the whole joint (angle-bars and rail) 
is only 52 4% of the strength of the unbroken rail. 

4. That the ineffectiveness of the angle-bar is due chiefly to 
a deficiency in compressive resistance. 

Although it has been universally recognized that the angle- 
bar is not a perfect form of joint, its simplicity, chea.pness, and 
reliability have caused its almost universal adoption. Within a 
very few years other forms (to be described later) have been 
adopted on trial sections and have been more and more extended, 
until their present use is very large. These designs all agree in 
using metal below the base oLihe rail, as is shown in the several 
designs on Plate VII, but the general type shown in Fig. 119 
is stiU (1912) in most common use. 

239. Effect of rail gap at joints. It has been found that the 
jar at a joint is due almost entirely to the deflection of the joint 
and scarcely at all to the small gap required for expansion. 
This gap causes a drop equal to the versed sine of the arc having 
a chord equal to the gap and a radius equal to the radius of 
the wheel. Taking the extreme case (for a 30-foot rail) of a f 
gap and a 33'' freight-car wheel, the drop is about t^Vtf''- ^^ 
order to test how much the jarring at a joint is due to a gap be- 
tween the rails, the experiment was tried of cutting shallow 
notches in the top of an otherwise solid rail and running a loco- 
motive and an inspection car over them. The resulting jarring 
was practically imperceptible and not comparable to the jar pro- 
duced at joints. Notwithstanding this fact, many plans have 

* Roadmasters Association of America — Reports for 1897. 



272 RAILROAD CONSTRUCTION. § 239. 

been tried for avoiding this gap. The most of these plans con- 
sist essentially of some form of compound rail, the sections 
breaking joints. (Of course the design of the compound rail 
has also several other objects in view.) In Fig. 117 are shown a 




Fig. 117. — Compound Rail Sections. 

few of the very many designs which have been proposed. These 
designs have invariably been abandoned after trial. Another 
plan, which has been extensively tried on the Lehigh Valley 
R. R., is the use of mitered joints. The advantages gained by 
their use are as yet doubtful, while the added expense is unques- 
tionable. The ^^ Roadmasters Association of America '' in 1895 
adopted a resolution recommending mitered joints for double 
track, but their use has been abandoned. 

240. " Supported," " suspended," and " bridge " joints. In a 
supported joint the ends of the rails are on a tie. If the angle- 
plates are short, the joint is entirely supported on one tie; if 
very long, it may be possible to place three ties under one angle- 
bar and thus the joint is virtually supported on three ties rather 
than one. In a suspended joint the ends of the rails are midway 
between two ties and the joint is supported by the two. There 
have always been advocates of both methods, but suspended 
joints are more generally used than supported joints. The 
opponents of three-tie joints claim that either the middle tie will 
be too strongly tamped, thus making it a supported joint, or 
that, if the middle tie is w^eakest, the joint becomes a very long 
(and therefore weak) suspended joint between the outer joint- 
ties, or that possibly one of the outer joint-ties gives way, thus 
breaking the angle-plate at the joint. Another objection which 
is urged is that unless the bars are very long (say 44 inches, as 
used on the Mich. Cent. R. R.) the ties are too close for proper 
tamping. The best answer to these objections is the successful 
use of these joints on several heavy-traffic roads 

"Bridge' '-joints are similar to suspended joints in that the 
joint is supported on two ties, but there is the important differ- 
ence that the bridge joint supports the rail from underneath and 



§ 242. RAIL-FASTENINGS. 273 

there is no transverse stress in the rail, whereas the suspended 
joint requires the combined transverse strength of both angle- 
bars and rail. A serious objection to bridge- joints Hes in the 
fact of their considerable thickness between the rail base and the 
tie. When joints are placed ^^ staggered " (as is now the invariable 
standard practice), rather than '^opposite,'' the ties support- 
ing a bridge-joint must either be notched down, thus weak- 
ening the tie and promoting decay at the cut, or else the tie 
must be laid on a slope and the joint and the opposite rail do not 
get a fair bearing. 

241. Failures of rail-joints. It has been observed on double- 
track roads that the maximum rail wear occurs a few inches 
beyond the rail gap at the joint in the direction of the traffic. 
On single-track roads the maximum rail wear is found a few 
inches each side of the joint rather than at the extreme ends of 
the rail, thus showing that the rail end deflects down under the 
wheel until (^\dth fast trains especially) the wheel actually jumps 
the space and strikes the rail a few inches beyond the joint, the 
impact producing excessive wear. This action, which is called 
the "drop," is apt to cause the first tie beyond the joint to 
become depressed, and unless this tie is carefully watched and 
maintained at its proper level, the stresses in the angle-bar may 
actually become reversed and-the bar may break at the top. The 
angle-bars of a suspended joint are normally in compression at 
the top. The mere reversal of the stresses would cause the bars 




Fig. 118. — Effect of "Wheel Drop" (Exaggerated). 

to give way with a less stress than if the stress were always the 
same in kind. A supported joint, and especially a three-tie 
joint (see § 240), is apt to be broken in the same manner. 

242. Standard angle-bars. An angle-bar must be so made 
as to closely fit the rails. The great multiplicity in the designs 
of rails (referred to in Chapter IX) results in nearly as great 
variety in the detailed dimensions of the angle-bars. The sec- 
tions here illustrated must be considered only as types of the 
variable forms necessary for each different shape of rail. The 



274 



RAILROAD CONSTRUCTION. 



§242. 



absolutely essential features required for a fit are (1) the angles 
of the upper and lower surfaces of the bar where they fit against 
the rail, and (2) the height of the bar. The bolt-holes in the 
bar and rail must also correspond. The holes in the angle-plates 
are elongated or made oval, so that the track-bolts, which are 




Fig. 119.— Standard Angle-bar— 80-lb. Rail. M. C. R.R. 



made of corresponding shape immediately under the head, will 
not be turned by jarring or vibration. The holes in the pails 
are made of larger diameter (by about A") than the bolts, so as 
to allow the rail to expand with temperature. 

The standard drilling for bolt-holes in the end of a rail, as 
vadopted by the A. R. E. & M. W. Assoc, in 1906, is as follows: 

End of rail to center first hole 23^ i^s. 

Center first hole to center second hole 5 

Center second hole to center third hole (for six-hole 

joint) ^ 

The proper length of angle-bars has not yet been standard- 
ized, but the above dimensions point very closely to the proper 
length. If the centers of the middle pair of holes in the angle- 
plate are made ^ inches apart, and the holes in the rails are 
A inch larger in diameter than the bolts, it will allow for an 
extreme variation in the length of the rails of f inch— due to 
expansion. Adding 4 inches at each end of the joint, from 



-fr^ 




CLOUD JOINT. 





ATLAS SUSPENDED RAIL JOINT. 



FISHER BRIDGE JOINT. 




r _ ^jj) 



^- 



WOLHAUPTER JOINT 



AfEBER RAIL JOINT. 



Plate VII. — Some forms of Rail Joints. 
{Between jip. 274 and 275.) 




BONZANQ RAIL JOINT, 



§ 243. RAIL-FASTENINGS. 275 

the center of the last hole to the end of the angle-plate, will 
make a length of 22| inches for a four-hole and 32f inches for 
a six-hole joint. This is considerably less than the M. C. R. R. 
joint shown above, but this joint was purposely lengthened so 
that it could be used for a three-tie joint. 

243. Later designs of rail-joints. In Plate VII are shown 
various designs which are competing for adoption. The most 
prominent of these (judging from the discussion in the conven- 
tion of the Roadmasters Association of America in 1897) are 
the ^'Continuous" and the ''Weber.'' Each of them has been 
very extensively adopted, and where used are universally pre- 
ferred to angle-plates. Nearly all the later designs embody 
more or less directly the principle of the bridge-joint, i.e., sup- 
port the rail from underneath. An experience of several years 
will be required to demonstrate which form of joint best satis- 
fies the somewhat opposed requirements of minimum cost (both 
initial and for maintenance) and minimum wear of rails and 
rolling stock. 

243a. Proposed specifications for steel splice-bars. The fol- 
lowing specifications for steel splice-bars were proposed in 1900 
by Committee No. 1, American Section, International Associa- 
tion for Testing Materials. 

1. Steel for splice-bars may^e made by the Bessemer or open- 
hearth process. 

2. Steel for splice-bars shall conform to the following limits 
in chemical composition: 

Per cent. 

Carbon shall not exceed 0.15 

Phosphorus shall not exceed 0.10 

Manganese 0.30 to 0.60 

3. SpKce-bar steel shaU conform to the following physical 
qualities : 

Tensile strength, pounds per square inch 54000 to 64000 

Yield point, pounds per square inch 32000 

Elongation, per cent in eight inches shall not 

be less than 25 

4. (a) A test specimen cut from the head of the splice-bar 
shall bend 180° flat on itself without fracture on the outside 
of the bent portion. 



276 RAILROAD CONSTRUCTION. § 243. 

(6) If preferred the bending test may be made on an un- 
punched splice-bar, which, if necessary, shall be first flattened 
and shall then be bent 180° flat on itself without fracture on 
the outside of the bent portion. 

5. A test specimen of 8-inch gauged length, cut from the head 
of the splice-bar, shall be used to determine the physical proper- 
ties specified in paragraph No. 3. 

6. One tensile specimen shall be taken from the rolled splice- 
bars of each blow or melt, but in case this develops flaws, or 
breaks outside of the middle third of its gauged length, it may 
be discarded and another test specimen submitted therefor. 

7. One test specimen cut from the head of the splice-bar shall 
be taken from a rolled bar of each blow or melt, or if preferred 
the bending test may be made on an unpunched splice-bar, 
which, if necessary, shall be flattened before testing. The bend- 
ing test may be made by pressure or by blows. 

8. For the purposes of this specification, the yield point shall 
be determined by the careful observation of the drop of the 
beam or halt in the gauge of the testing machine. 

9. In order to determine if the material conforms to the chem- 
ical limitations prescribed in paragraph No. 2 herein, analysis 
shall be made of drillings taken from a small test ingot. 

10. All splice-bars shall be smoothly rolled and true to templet. 
The bars shall be sheared accuratel}'- to length and free from 
fins or cracks, and shall perfectly fit the rails for which they are 
intended. The punching and notching shall accurately conform 
in every respect to the drawing and dimensions furnished. 

11. The name of the maker and the year of manufacture shall 
be rolled in raised letters on the side of the splice-bar. 

12. The inspector representing the purchaser shall have all 
reasonable facilities afforded to him by the manufacturer, to 
satisfy him that the finished material is furnished in accordance 
with these specifications. All tests and inspections shall be 
made at the place of manufacture, prior to shipment. 

TIE-PLATES. 

244. Advantages, (a) As already indicated in § 204, the 
life of a soft-wood tie is very much reduced by '' rail-cutting'' 
and ^^spike-killing," such ties frequently requiring renewal long 



§ 244. RAIL-FASTENINGS. 277 

before any serious decay has set in. It has been practically 
demonstrated that the "rail-cutting'^ is not due to the mere 
pressure of the rail on the tie, even with a maximum load on 
the rail, but is due to the impact resulting from vibration and 
to the longitudinal working of the rail. It has been proved 
that this rail-cutting is practically prevented by the use of tie- 
plates, (h) On curves there is a tendency to overturn the outer 
rail due to the lateral pressure on the side of the head. 
This produces a concentrated pressure of the outer edge of the 
base on the tie which produces rail-cutting and also draws the 
inner spikes. Formerly the only method of guarding against 
this was by the use of "rail-braces/' one pattern of which is 
shown in Fig. 120. But h nas been found that tie-plates serve 
the purpose even better, and rail -braces have been abandoned 
where tie-plates are used, (c) Driving spikes through holes 
in the plate enables the spikes on each side of the rail to mutually 
support each other, no matter in which (lateral) direction the 
rail may tend to move, and this probably accounts in large 




TFio. 120. — Atlas Brace K, 

measure for the added stability obtained by the use of tie-plates. 

(d) The wear in spikes, called "necking," caused by the ver- 
tical vibration of the rail against them, is very greatly reduced. 

(e) The cost is very small compared with the value of the added 
life of the tie, the large reduction in the work of track main- 
tenance, and the smoother running on the better track which is 
obtained. It has been estimated that by the use of tie-plates 
the life of hard-wood ties is increased from one to three years, 
and the life of soft-wood ties is increased from three to six 
years. From the very nature of the case, the value of tie-plates 
is greater when they are used to protect soft ties. 



27S 



■RAILROAD CONSTRUCTION. 



§245. 



245. Elements of the design. There is still a great diversity 
of opinion regarding the relative advantages of tie-plates which 
are flat on the bottom and those which are corrugated or pro- 
vided with teeth or claws which are imbedded in the tie. Those 
used in Europe are without exception flat on the bottom. The 
Pennsylvania Railroad and the Southern Pacific have also 
been using flat tie-plates. On the other hand, it is claimed 
I that the pressure required to force a corrugated plate into a 
Uie is about 20% greater than that required to imbed a flat 




Wolhaupter 





Round grqoved-tapered-flat 
bottom-shoulder tie plate 



P.R.R. flat bottom tie plate 
Claw and shoulder tie plate 

Fig. 121. — Various Forms of Tie-plates. 



plate of equal thickness in the tie. It is also claimed that tests 
have shown that the force required to spread the rails when 
they are fastened with corrugated plates under the rails is 
from 36% to 100% greater than that required when a flat 
tie-plate is used. It is especially important that the plate 
shall be so firmly imbedded in the tie that it cannot move or 
''rock" with each motion of the rail over it. Instances are 
known where a treated tie has become unfit for service because 



§ 245. RAIL-FASTENINGS. 279 

the tie-plate has rocked back and forth until it has worn a 
hole in the tie. Rain-water filling this hole has leached out 
the zinc chloride, and the tie has decayed at this point and 
become unserviceable, when the remainder of the tie showed 
no decay. The creeping of the rails over the ties is sometimes 
the cause of failure of ties which have been effectually secured 
against decay by the use of preservatives. This particular form 
of tie deterioration has been guarded against on a French rail- 
road by using a tie-plate made of creosoted wood, w^hich is 8 
inches long, the same width as the width of the base of the rail, 
and J inch thick. Such wooden plates, which will last a year 
and a half to two years, are made of poplar, or any other hard 
wood, and cost about 12.00 per thousand. It should be ob- 
served that they are used in connection with wooden screws 
instead of the ordinary track spikes. When they are worn 
out, it is only necessary to turn the scr:^w one or two upward 
turns ; the new plate may then be put in endwise and the screw- 
spikes again fastened down. 

A fault of the earlier designs of metal tie-plates was that 
they were made of plates which were so thin that they would 
buckle under the pressure of the railo The claim made for the 
corrugated plates is that their transverse stress is far greater 
than that of a flat plate for the same amount of material ; but 
this is not vital provided the flat plates are made sufficiently 
thick so that they will not buckle. The tie-plate used on the 
Southern Pacific Railroad has a slightly beveled surface, the 
plate being | inch thick under the outer edge of the rail and 
^1^ inch thick at the inner edge of the rail. 

The holes in a tie-plate should be about -^^ inch larger than 
the size of the intended spike. For example, the holes are 
generally punched with f X f" holes for a y^ inch spike, or 
with holes | inch square for a J inch spike. The length of the 
plate (perpendicular to the rail) should be about 3 inches more 
than the base of the rail; this usually means about 8 inches* 
For very heavy traffic, the thickness should be j\ inch to f inch ; 
for average traffic it may be as thin as J inch; some plates are 
made only -^-^ inch thick. For flat-bottom plates the thick- 
ness may be as great as half an inch. The tie-plates under tha 
joint ties must be somewhat longer than the intermediates, in order 
to allow for the extra length from out to out of the angle- 
plates. 



280 RAILROAD CONSTRUCTION. § 246. 

246. Method of setting. A very important detail in the 
process of setting the tie-plates on the ties is that the plates 
should be rigidly attached to the ties in their intended position 
during the process of setting. If tie-plates with flat bottoms 
are used, the surface of the tie must be adzed, so that it is not 
only plane but level, so that there will be no danger that the 
plate will rock on the tie. When using tie-plates which are 
corrugated on the under surface, it is necessary to force them 
into the tie until the under side of the plate is flush with the 
surface of the tie. This requires a pressure of several thousand 
pounds. Sometimes trackmen have depended on the easy 
process of waiting for passing trains to force the corrugations 
into the tie until the plate is in its intended position. Until 
the plates are finally set the spikes cannot be driven home, 
and this apparently cheap and easy process generally results 
in loose spikes and rails. The best method for new work is 
to drive the plates into the tie before setting the tie in position. 
A tie-plate gauge holds both tie-plates in their proper relative 
position, and both plates may be driven by the use of heavy 
beetles. When it is necessary to place the plate under the rail 
^nd drive it in, it is somewhat difficult to drive it by striking 
the plate with a swage on each side of the rail alternately. 
When it is struck on one side, the other side flies up unless held 
down by a wedge driven between the plate and the rail on the 
other side of the rail. A straddler, which straddles the rail 
somewhat like an inverted U, is very useful for this purpose, 
since it makes it possible to strike the head of the straddler and 
force down both sides of the plate at once. The Southern 
Pacific Railroad Company has rigged up a small pile-driver on 
a hand-car, which is used in connection with a straddler to drive 
the tie-plates into position. Some western railroads have even 
adopted the process of rigging up a flat car with a machine 
which will press the tie-plates into place in the ties before the 
ties are placed in the track. 

SPIKES. 

247. Requirements. The rails must be held to the ties by a 
fastening which will not only give sufficient resistance, but which 
will retain its capacity for resistance. It must also be cheap 
^nd easily applied. The ordinary track-spike fulfills the last 



§247. 



RAIL-FASTENINGS. 



281 



requirements, but has comparatively small resisting power, com- 
pared with screws or bolts. Worse than all, the tendency to 






Fig. 122. 



Fig. 123. 



vertical vibration in the rail produces a series of upward pulls on 
the spike that soon loosens it. When motion has once begun 
the capacity for resistance is greatly reduced, and but little more 
vibration is required to pull the spike out so much that redriving 
is necessary. Driving the spike to place again in the same hole 
is of small value except as a very temporary expedient, as its 
holding power is then very small. Redriving the spikes in new 
holes very soon ^' spike-kills" the tie. Many plans have been 
devised to increase the holding power of spikes, such as making 
them jagged, twisting the spike, swelling the spike at about the 
center of its length, etc. But it has been easily demonstrated 
that the fibers of the wood are generally so crushed and torn by 
driving such spikes that their holding powxr is less than that of 
the plain spike. 

The ordinary spike (see Fig. 122) is made with a square cross- 
section which is uniform through the middle of its length, the 
lower If" tapering dowTi to a chisel edge, the upper part swelling 
out to the head. The Goldie spike (see Fig. 123) aims to im- 
prove this form by reducing to a minimum the destruction of the 
fibers. To this end, the sides are made smooth, the edges are 
clean-cut, and the point, instead of being chisel-shaped, is ground 



282 



RAILROAD CONSTRUCTION. 



§ 248. 



down to a pyramidal form. Such fiber-cutting as occurs is thus 
accomphshed without much crushing, and the fibers are thus 
pressed away from the spike and shghtly downward. Any 
tendency to draw the spike will therefore cause the fibers to 
press still harder on the spike and thus increase the resistance. 

248. Driving. The holding power of a spike depends largely 
on how it is driven. If the blows 
are eccentric and irregular in direc- 
tion, the hole will be somewhat en" 
larged and the holding power largely 
decreased. The spikes on each 
side of the rail in any one tie should 
not be directly opposite, but should 
be staggered Placing them direct- 
ly opposite will tend to split the tie, 
or at least decrease the holding 
power of the spikes. The direction 
of staggering should be reversed in 
the two pairs of spikes in any one 
tie (see Fig. 124), This will tend to prevent any twisting of the 
tie in the ballast, which would otherwise loosen the rail from the 
tie. 

249. Screws and bolts. The use of these abroad is very ex- 
tensive, but their use in this country has not passed the experi- 
mental stage. The screws are "wood "-screws (see Fig. 125), 



Fig. 124. — Spike-driving. 




Fig. 125. — Screw Spike. 



§ 250. RAIL-FASTENINGS. 283 

having large square heads, which are screwed down with a track- 
wrench. Holes, having the same diameter as the base of the 
screw-heads, should be first- bored into the tie, at exactly the 
right position and at the proper angle with the vertical. A 
light wooden frame is sometimes used to guide the auger at the 
proper angle. Sometinies the large head of the screw bears 
directly against the base of the rail, as with the ordinary spike. 
Other designs employ a plate, made to fit the rail on one side, 
bearing on the tie on the other side, and through which the screw 
passes. These screws cost much more than the spikes and re- 
quire more work to put in place, but their holding power is much 
greater and the work of track maintenance is very much less. 
Screw-bolts, passing entirely through the tie, ha^dng the head 
at the bottom of the tie and the nut on the upper side, are also 
used abroad. These are quite difficult to replace, requiring that 
the ballast be dug out beneath the tie, but on the other hand the 




Fig. 126. 
occasions for replacing such a bolt are comparatively rare, a» 
their durability is very great. The use of screws or bolts in- 
creases the life of the tie by the avoidance of '' spike-kiUing.'' It 
is capable of demonstration that the reduced cost of mainte- 
nance and the resulting improvement in track would much more 
than repay the added cost of screws and bolts, but it seems im- 
possible to induce railroad directors to authorize a large and 
immediate additional expenditure to make an annual saving 
whose value, although unquestionably considerable, cannot.be 
exactly computed. 

250. "Wooden spikes." Among the regulations for track- 
laying given in § 208, mention was made of wooden " spikes," 



284 



RAILROAD CONSTRUCTION. 



§250. 



r'liu 



or plugs, which are used to fill up the holes when spikes are 
withdrawn. The value of the policy of filhng up these holes is 
unquestionable, since the expense is insignificant compared with 
the loss due to the quick and certain decay of the tie if these 
holes are allowed to fill with water and remain so. But the 
method of making these plugs is variable. On some roads they 
are "hand-made'' by the trackmen out of otherwise use- 
less scraps of lumber, the work being done at odd mo- 
ments. This policy, while apparently cheap, is not 
necessarily so, for the hand-made plugs are irregular 
in size and therefore more or less inefficient. It is 
also quite probable that if the trackmen are required to 
make their own plugs, they would spend time on these 
very cheap articles which could be more profitably em- 
ployed otherwise. Since the holes made by the spikes 
are larger at the top than they are near the bottom, the 
plugs should not be of uniform cross-section but should 
be slightly wedge-shaped. The "Goldie tie-plug" 
(see Fig. 127) has been designed to fill these require- 
ments. Being machine-made, they are uniform in 
size; they are of a shape which will best fit the hole; 
they can be furnished of any desired wood, and at a 
cost which makes it a wasteful economy to attempt 
to cut them by hand. 



n; 



Fig. 127. 



TRACK-BOLTS AND NUT- LOCKS. 

251. Essential requirements. The track-bolts must have 
sufficient strength and must be screwed up tight enough to hold 
the angle-plates against the rail with sufficient force to develop 
the full transverse strength of the angle-bars. On the other 
hand the bolts should not be screwed so tight that slipping may 
not take place when the rail expands or contracts with tempera- 
ture. It would be impossible to screw the bolts tight enough to 
prevent slipping during the contraction due to a considerable fall 
of temperature on a straight track, but when the track is curved, 
or when expansion takes place, it is conceivable that the resist- 
ance of the ties in the ballast to lateral motion may be less than 
the resistance at the joint. A test to determine this resistance 
was made by Mr. A. Torrey, chief engineer of the Mich. Gent. 
R. R., using 80-lb. rails and ordinary angle-bars, the bolts being 
screwed jip as usual. If required a force of about 31000 to 



§ 252. 



RAIL-FASTENINGS. 



285 



35000 lbs. to start the joint, which would be equivalent to the 
stress induced by a change of temperature of about 22°. But 
if the central angle of any given curve is small, a comparatively 
small lateral component will be sufficient to resist a compression 
of even 35000 lbs. in the rails. Therefore there will ordinarily 
be no trouble about having the joints screwed too tight. The 
vibration caused by the passage of a train reduces the resistance 
to slipping. This vibration also facilitates an objectionable 
feature, viz., loosening of the nuts of the track-bolts. The bolt 
is readily prevented from turning by giving it a form which is 
not circular immediately under the head and making corre- 
sponding holes in the angle-plate. Square holes would answer 
the purpose, except that the square corners in the holes in the 
angle-plates would increase the danger of fracture of the plates. 
Therefore the holes (and also the bolts, under the head) are 
made of an oval form, or perhaps a square form with rounded 
corners, avoiding angles in the outline. 

The nut-locks should be simple and cheap, should have a life 
at least as long as the bolt, should be effective, and should not 
lose their effectiveness Tvdth age. Many of the designs that have 
been tried have been failures in one or more of these particulars 
as will be described in detail below. 

252. Design of track-bolts. - In Fig. 128 is shown a common 
design of track-bolt. In its general form this represents the 

bolt used on nearly all roads, 
being used not only with the 
common angle-plates, but also 
with many of the improved de- 
signs of rail- joints. The varia- 
tions are chiefly a general in- 
crease in size to correspond with 
the increased weight of rails, 
besides variations in detail di- 
mensions which are frequently 
unimportant. The diameter is 
usually I" to f''; 1" bolts are 
sometimes used for the heaviest 
sections of rails. As to length, 

the bolt should not extend more 
Fig. 128.-TRACK-BOLT. ^^^^ ^„ ^^^^.^^ ^^ ^^^ ^^^ ^^^^ 

it is screwed up. If it extends farther than this it is liable to be 




286 RAILROAD CONSTRUCTION. § 253. 

broken off by a possible derailment at that point. The lengths 
used vary from 3^", which may be used with 60-lb. rails, to 5", 
which is required with 100-lb. rails. The length required de- 
pends somewhat on the type of nut-lock used. 

253. Design of nut-locks. The designs for nut-locks may be 
divided into three classes: (a) those depending entirely on an 
elastic washer which absorbs the vibration which might other- 
wise induce turning; (b) those which jam the threads of the 
bolt and nut so that, when screwed up, the frictional resistance 
is too great to be overcome by vibration; (c) the "positive" 
nut-locks — those which mechanically hold the nut from turning. 
Some of the designs combine these principles to some extent. 
The "vulcanized fiber'' nut-lock is an example of the first class. 
It consists essentially of a rubber washer which is protected by 
an iron ring. When first placed this lock is effective, but the 
rubber soon hardens and loses its elasticity and it is then ineffec- 
tive and worthless. Another illustration of class (a) is the use 
of wooden blocks, generally 1" to 2'' oak, which extend the 
entire length of the angle-bar, a single piece forming the washer 
for the four or six bolts of a joint. This form is cheap, but the 
wood soon shrinks, loses its elasticity, or decays so that it soon 
becomes worthless, and it requires constant adjustment to keep 
it in even tolerable condition. The "'Verona'' nut-lock is 
another illustration of class (a) which also combines some of the 
positive elements of class (c). It is made of tempered steel and, 
as shown in Fig. 129, is warped and has sharp edges or points. 
The warped form furnishes the element of elastic pressure when 
the nut is screwed up. The steel being harder than the iron of 
the angle-bar or of the nut, it bites into them, owing to the 
great pressure that must exist when the washer is squeezed 
nearly flat, and thus prevents any backward movement, although 
forward movement (or tightening the bolt) is not interfered 
with. The " National" nut-lock is a type of the second class (h), 
in which, like the " Harvey" nut-lock, the nut and lock are com- 
bined in one piece. With six-bolt angle-bars and 30-foot rails, 
this means a saving of 2112 pieces on each mile of single track. 
The "National" nuts are open on one side. The hole is drilled 
and the thread is cut slightly smaller than the bolt, so that when 
the nut is screwed up it is forced slightly open and therefore 
presses on the threads of the bolt with such force that vibration 
cannot jar it loose. Unlike the " National " nut, the '' Harvey " 



§253. 



RAIL-FASTENINGS. 



287 





VERONA 



VULCANIZED FIBRE 




IMPROVED VERONA 




NATIONAL 




Columbia.Nut Lock 



C 




JONES 



Fis. 123. — ^Types op Nut-locks. 



288 RAILROAD CONSTRUCTION. § 253. 

nut is solid, but the form of the thread is progressively varied so 
that the thread pinches the thread of the bolt and the frictional 
resistance to turning is too great to be affected by vibration. 

The ^'Columbia'' nut-lock is a two-piece nut, both parts of 
which must turn simultaneously. As shown in the figure, one 
section wedges into the other. The greater the tension in the 
bolt, the greater the wedging action and the greater the friction 
to prevent turning. 

The "Jones'* nut-lock, belonging to class (c), is a type of a 
nut-lock that does not depend on elasticity or jamming of screw- 
threads. It is made of a thin flexible plate, the square part of 
which is so large that it will not turn after being placed on the 
bolt. After the nut is screwed up, the thin plate is bent over so 
that the re-entrant angle of the plate engages the corner of the 
nut and thus mechanically prevents any turning. The metal 
is supposed to be sufficiently tough to endure without fracture 
as many bendings of the plate as will ever be desired. Nut- 
locks of class (c) are not in common use. 



CHAPTER XI. 

SWITCHES AND CROSSINGS. 

SWITCH CONSTRUCTION. 

254. Essential elements of a switch. Flanges of some sort are 
a necessity to prevent car-wheels from ninning off from the rails 
on which they may be moving. But the flanges, although a 
necessity, are also a source of comphcation in that they require 
some special mechanism which mil, when desired, guide the 
wheels out froni the controlling influence of the main-line rails. 
This must either be done by raising the wheels high enough 
so that the flanges may pass over the rails, or by breaking the 
continuity of the rails in such a waj'^ that channels or "flange 
spaces" are formed through the rails. An ordinary stub-switch 
breaks the continuity of the main-line rails in three places, two 
of them at the switch-block and one at the frog. The Wharton 
switch avoids two of these breaks by so placing inclined planes 
that the wheels, rolling on their flanges, will surmount these 
inclines until they are a little higher than the rails. Then the 
wheels on the side toward which the switch runs are guided 
over and across the main rail on that side This rise being ac- 
complished in a short distance, it becomes impracticable to 
operate these s\^dtches except at slow speeds, as any sudden 
change in the path of the center of gravity of a car causes very 
destructive jars both to the STvitch and to the rolling stock. The 
other general method makes a break in one main rail (or both) 
at the switch-block. In both methods the wheels are led to one 
side by means of the "lead rails," and finally one line of wheels 
passes through the main rail on that side by means of a "frog." 
There are some designs by which even this break in the main 
rail is avoided, the wheels being led over the main rail by means 
of a short movable rail which is on occasion placed across the 
main rail, but such designs have not come into general use. 

255. Frogs. Frogs are provided with two channel-ways or 
"flange spaces" through which the flanges of the wheels move, 

289 



290 



RAILROAD CONSTRUCTION. 



§255. 



Each channel cuts out a parallelogram from the tread area. 
Since the wheel-tread is always wider than the rail, the wing 
rails will support the wheel not only across the space cut out by 
the channel, but also until the tread has passed the point of the 
frog and can obtain a broad area of contact on the tongue of the 
frog. This is the theoretical idea, but it is very imperfectly 




Fig. 130. — Diagrammatic Design of Frog. 



realized. The wing rails are sometimes subjected to excessive 
wear owing to '^hollow treads'' on the wheels — owing also to 
the frog being so flexible that the point ^' ducks" when the wheel 
approaches it. On the other hand the sharp point of the frog 
will sometimes cause destructive wear on the tread of the wheel. 
Therefore the tongue of the frog is not carried out to the sharp 
theoretical point, but is purposely somewhat blunted. But 
the break which these channels make in the continuity of the 
tread area becomes extremely objectionable at high speeds, 
being mutually destructive to the rolling stock and to the frog. 
The jarring has been materially reduced by the device of '^spring 
frogs" — to be described later. Frogs were originally made of 
cast iron — then of cast iron with wearing parts of cast steel, 
which were fitted into suitable notches in the cast iron. This 
form proved extremely heavy and devoid of that elasticity of 
track which is necessary for the safety of rolling stock and 
track at high speeds. The present universal practice is to build 
the frog up of pieces of rails which are cut or bent as required. 
These pieces of rails (at least four) are sometimes assembled by 
riveting them to a flat plate, but this method is now but little 
used, except for very light work. The usual practice is now 
chiefly divided between "bolted" and "keyed" frogs. In each 
case the space between the rails, except a sufficient flange-way, 
is filled with a cast-iron filler and the whole assemblage of parts 



§ 255, 



SWITCHES AND CROSSINGS. 



291 




Plate VIII. — Some Types of Frogs. 



292 RAILROAD CONSTRUCTION. § 255. 

is suitably bolted or clamped together, as is illustrated in Plate 
VIII. The operation of a spring-rail frog is evident from the 
figure. Since a siding is usually operated at slow speed, while 
the main track may be operated at fast speed, a spring-rail frog 
will be so set that the tread is continuous for the main track and 
broken for the siding. This also means that the spring-rail will 
only be moved by trains moving at a (presumably) slow speed 
on to the siding. For the fast trains on the main line such a 
frog is substantially a ^' fixed ^' frog and has a tread which is 
practically continuous. 

256. To find the frog number. The frog number (n) equals 
the ratio of the distance of any point on the tongue of the frog 
from the theoretical point of the frog divided by the width of 
the tongue at that point, i.e. =hc-^ah (Fig. 130). This value 
may be directly measured by applying any convenient unit of 
measure (even a knife, a short pencil, etc.) to some point of the 
tongue where the width just equals the unit of measure, and then 
noting how many times the unit of measure is contained in the 
distance from that place to the theoretical point. But since c, 
the theoretical point, is not so readily determinable with exacti- 
tude, it being the imaginary intersection of the gauge lines, it 
may be more accurate to measure de, ah, and hs; then n, the frog 
number, =hs-~ (ah + de) . If the frog angle be called F, then 

n=hc-~ah=hs-i-{ah + de)=^ cot ^F; 
i.e., cot iF=2n. 

257. Stub switches. The use of these, although once nearly 
universal, has been practically abandoned as turnouts from 
main track except for the poorest and cheapest roads. In some 
States their use on main track is prohibited by law. They have 
the sole merit of cheapness with adaptability to the circum- 
stances of very light traffic operated at slow speed when a con- 
siderable element of danger may be tolerated for the sake of 
economy. The rails from A to B (see Fig. 131 *) are not fastened 

*The student should at once appreciate that in Fig. 131, as well as in 
nearly all the remaining figures in this chapter, it becomes necessary to use 
excessively large frog angles, short radii, and a very wide gauge in order to 
illustrate the desired principles with figures which are sufficiently small for 
the page. In fact, the proportions used in the figures are such that serious 
mechanical difficulties would be encountered if they were used. These dif- 
ficulties are here ignored because they can be neglected in the proportions 
used in practice. 



I 257. 



SWITCHES AND CROSSINGS. 



293 



to the ties; they are fastened to each other by tie-rods which 
keep them at the proper gauge; at and back of B they are 




Fig. 131. — Stub Switch. 

securely spiked to the ties, and at A they are kept in place by 
the connecting bar ((7) fastened to the switch-stand. One great 
objection to the switch is that, in its usual form, when operated 
as a trailing switch, a derailment is inevitable if the switch is 
misplaced. The very least damage resulting from such a derail- 
ment must include the bending or breaking of the tie-rods of the 
s\\dtch-rail. Several devices have been invented to obviate this 
objection, some of which succeed very well mechanically, al- 
though their added cost precludes any economy in the total cost 
of the switch. Another objection to the switch is the looseness 
of construction which makes the switches objectionable at high 
speeds. The gap of the rails at the head-block is always con- 
siderable, and is sometimes as much as two inches. A driving- 




FiG. 132, — Point Switch. 



wheel with a load of 12000 to 20000 pounds, jumping this gap 
with any considerable velocity, will do immense damage to the 



294 



BAILROAD CONSTRUCTION. 



§ 258. 



farther rail end, besides producing such a stress in the construc- 
tion that a breakage is rendered quite Hkely, and such a breakage 
might have very serious consequences. 

258. Point switches. The essential principle of a point switch 
is illustrated in Fig. 132. As is shown, one main rail and also 
one of the switch-rails is unbroken and immovable. The other 
main rail (from A to F) and the corresponding portion of the 
other lead rail are substantially the same as in a stub switch. 
A portion of the main rail (AB) and an equal length of the oppo- 
site lead rail (usually 15 to 24 feet long) are fastened together 
by tie-rods. The end at A is jointed as usual and the other end 
is pointed, both sides being trimmed down so that the feather 
edge at B includes the web of the rail. In order to retain in it 
as much strength as possible, the point-rail 
is raised so that it rests on the base of the 
stock-rail, one side of the base of the 
point-rail being entirely cut away. As [ 
may be seen in Fig. 133, although the in- \ 
fluence of the point of the rail in moving 
the wheel-flange away from the stock-rail 
is really zero at that point, yet the rail has 
all the strength of the web and about one- 
half that of the base — a very fair angle- 
iron. The planing runs back in straight 
lines, until at about six or seven feet back 
from the point the full width of the head is 
obtained. The full width of the base will only be obtained at 
about 13 feet from the point. The A. H. E. A. standard switch 
rail is always cut on the basis that the distance between 
gauge lines at the heel of the switch (the distance MN in 
Fig. 140) is 6i inches and that the ^^point^' is J inch wide. 
Then, using four standard lengths, 11, 16i, 22, and 33 feet, 
the angles vary from 2° 36' 19" to 0° 52' 05", as shown in 
Table III. 




Fig. 133. 



4 



259. Switch-stands. The simplest and cheapest form is the 
'*' ground lever, '^ which has no target. The radius of the circle 
described by the connecting-rod pin is precisely one-half the 
throw. From the nature of the motion the device is practically 



260. 



SWITCHES AND CROSSINGS. 



295 



self-locking in either position, padlocks being only used to pre- 
vent malicious tampering. The numerous designs of upright 
stands are always combined with targets, one design of which is 





Fia. 



134. — Ground Lever for Throwing 
A Switch. 



¥iQ. 135. 



illustrated in Fig. 185, When the road is equipped with inter- 
locking signals, the switch-throw mechanism forms a part of the 
design 

260. Tie-rods. These are fastened to the webs of the rails by 
means of lugs which are bolted on, there being usually a hinge- 
joint between the rod and the lug. Four such tie-rods ar@ 



296 



RAILROAD CONSTRUCTION. 



260. 



generally necessary. The first rod is sometimes made with- 
out hinges, which gives additional stiffness to the comparatively 
weak rail-points. The old-fashioned tie-rod, having jaws 
fitting the base of the rail, was almost universally used in the 
days of stub switches. One great inconvenience in their use 
lies in the fact that they must be slipped on, one by one, over 
the free ends of the switch-rails. 



,SL 



ja 



^ 



;h s_ 



ir=r Tai iSr 



(m 






M 



LifL 



ja—fij 



~ir 



^ar 



o 



J - 



-— - 2 



261. Guard-rails. As shown in Figs. 131 and 132, guard-rails 
are used on both the main and switch tracks opposite the frog- 
point. Their function is not only to prevent the possibility of 



§262. 



SWITCHES AND CROSSINGS. 



297 



the wheel-flanges passing on the wrong side of the frog-point, 
but also to save the side of the frog-tongue from excessive wear. 
The flange-way space between the heads of the guard-rail and 
wheel-rail should equal If inches. Since this is less than the 
space between the heads of ordinary (say 80-pound) rails when 
placed base to base, to say nothing of the |" required for spikes, 
it becomes necessary to cut the flange of the guard-rail. The 
length of the rail should be 16 feet 6 inches, the middle portion 
being straight for a length of 3 feet 6 inches, and the ends, each 
being 6 feet 6 inches long, curved out so that the side of the 
rail head at each end is 4 inches from the main rail head, when 
the flange-way at the center is If inches. See Fig. 136a 



MATHEMATICAL DESIGN OF STVITCHES. 

In all of the following demonstrations regarding switches, 
turnouts, and crossovers, the lines are assumed to represent the 
gauge-lines — i.e., the lines of the inside of the head of the rails. 
262. Design with circular lead-rails. The simplest method 

is to consider that the lead-rails 
curv^e out from the main track- 
rails by arcs of circles which are 
tangent to the main rails and 
which extend to the frog-point F. 
The simple curve frpm D to F is 
of such radius that (r + ig) vers F 
=g, in which i^ = the frog angle, 
g=gauge, L=the ''lead" (BF), 
and r=the radius of the center of 
the switch-rails. 




Fig. 137. 



r-hig = 



vers F' 



(74) 



Also, 
Also, 



BF^BD==cotiF', BD=g; BF=L. 

.-. L=g cot iF (75) 

L = (r + ig)smF; ..... (76) 

QT=2rsmiF. ........ (77) 



These formulae involve the angle F. As shown in Table III, 
the angles (F) are always odd quantities, and their trigonometric 
functions are somewhat troublesome to obtain closely with 



298 RAILROAD CONSTRUCTION. § 262. 

ordinary tables. The formul?? may be simplified by substitut- 
ing the frog-number n, from the relation that n = | cot ^F, 
Since 

r — ig=L cot F and r-{-ig=L cosec F, 

then r = JL (cot F + cosec F) 

= ig ^ot §i^(cot i^'^- cosec F) 

= hg cot^ Ji^, since (cot a + cosec a) =cot Ja 

= 2gn' , , . (78) 

Also, L = 2gn, (79) 

from which r=nXL. . . , (80) 

These extremely simple relations may obviate altogether the 
necessity for tables, since they involve only the frog-number and 
the gauge. On account of the great simphcity of these rules, 
they are frequently used as they are, regardless of the fact that 
the curve is never a uniform simple curve from switch- block to 
frog. In the first place there is a considerable length of the 
gauge-line within the frog, which is straight unless it is pur- 
posely curved to the proper curve while being manufactured, 
which is seldom if ever done — except for the very large-angled 
frogs used for street-railway work, etc. It is'also doubtful whether 
the smtch-rails (BA, Fig. 131) are bent to the computed curve 
when the rails are set for the switch. The switch-rails of point 
switches are straight, thus introducing a stretch of straight track 
which is about one-fifth of the total length of the lead-rails. The 
effect of these modifications on the length and radius of the lead- 
rails will be developed and discussed in the next four sections. 

The throw {t) of a stub switch depends on the weight of the 
rail, or rather on the width of its base. The throw must be at 
least J" more than that width. The head-block should there- 
fore be placed at such a distance from the heel of the switch {E) 
that the versed sine of the arc equals the throw. These points 
must be opposite on the two rails, but the points on the two rails 
where these relations are exactly true will not be opposite. 
Therefore, instead of considering either of the two radii {r-{-\g) 
and (r— Jgf), the mean radius r is used. Then (see Fig. 137) 

vers KOQ=t^r, 

and the length of the switch-rails is 

QK=r sin KOQ ' . . (81) 



§264. 



SWITCHES AND CROSSINGS. 



299 



Stub-switclies are generally used witli large frog angles. For 
small frog angles (large frog-numbers) the values of QK are so 
great that the length of rail left unspiked is too great for a safe 
track. If this were obviated by spiking down a portion of the 
lead the theoretical accuracy of the switch would be lost. The 
values of QK for various frog-numbers is given in Table III, 
These are based on a uniform throw {t) of 5^ inches. 

263. Effect of straight frog-rails. A portion of the ends of 

the rails of a frog arg free and may 
be bent to conform to the switch- 
rail curve, but there is a consid- 
erable portion which is fitted to 
the cast-iron filler, and this por- 
tion is always straight. Call the 
length of this straight portion 
back from the frog-point / ( =FH, 
Fig. 138). Then we have 
r + hg = {g-f sin F) -^vers F 

=-^-fcotiF 

vers F ' 



/''' FH=/ \ 


F 


t B 





Fig. 138. 



g 



\eTs F 



-2fn. 



(82) 



BF^L = {g-f sin F) cot ^F+f cos F 
=2gn-f sin F cot iF-^f eos F 
=2firn-/(l -f cos F) +/ cos F 
-=2gn-f 



(83) 



Since r—\g = {L—f sec F) cot F^ and 
r + ig = (L—f cos F) cosec F, 
r=iL(cot i^^-h cosec F)—y sec F cot F — y cos F cosec F 



^' \ sm/ / 



r=L?i — 5/cotii^ 

=^Ln-fn, Then from (83) 
r=2sfn2-2/n. ..... 



(84) 



264, Effect of straight point-rails. The "point switches,'' 
now so generally used, have straight switch-rails. This requires 



300 



KAILROAD CONSTRUCTION. 



§264. 



an angle in the alignment rather than turning off by a tangential 
curve. The angle is, however, very small (between 1° and 3°), 
and the disadvantages of this angle are small compared with 
the very great advantages of the device. 




^'^=shr«i^ + a)' 



r+i^=^-r 



FM 



2smh(F—a) 
g-k 



'2smi{F + a)smi(F-a) 
g-k 



1 



cos a — cos F' 

BF^L= FM cos ^{F + a) + DN 
^(g-k)cotKF-ha)+DN. . 



(85) 

(86) 



265. Combined effect of straight frog-rails and straight point- 
rails. It becomes necessary in this case to find a curve which 
shall be tangent to both the point-rail and the frog-rail. The 
curve therefore begins at M, its tangent making an angle of a 
(varying from 0° 52' to 2° 36') with the main rail, and runs to H. 



§ 266. 



SWITCHES AND CROSSINGS. 



301 



The central angle of the curve is therefore (F—d). The angle 
of the chord HM with the main rails is therefore 

i(F-a)+a=iiF + a); 
sinK-^ + a) ' 



r+i9' = 



HM 



2sini{F-a) 

g— f sin F—k 
" 2 sin i(F + a) sin i(F--a) 
_g— f sin F—k ^ 



cos a —cos i^' 

ST =2r sin i(F -a). 



(87) 

(88) 



BF=L=HM cos i(F + a)-{-f cos F+DN 

= (g-f sin F-k) cot i(F + a) +/ cos F^DN. . . (89) 

It may be more simple, if (r + Jg) has already been computed, 
to write 

L=2(r4-J9'} sini(^-a) cos ^{F + a) -\-f cos F-^-DN 
= (^ + hg) (sin i^ -sin a) +J cos F+DN (90) 



h-n 




Fig. 140. 



266. Comparison of the above methods. Computing values 
for r and L by the various methods, on the uniform basis of a 



302 



RAILROAD CONSTRUCTION. 



266. 



No. 9 frog, standard gauge 4' 8|", /=6'.00, /b = 6J"=0'.521, 
DN = 16' 6", and a = 1° 44' 11", we may tabulate the comparative 
results: 



§ 262. 
Simple circle. 
Curved frog- 
rail. Curved 
switch-rail. 



§263. 

Straight 

frog-rail. 

Curved 

switch-rail. 



§264. 
Curved frog- 
rail. Straight 
switch-rail. 



§ 265. 
Straight 
frog-rail. 
Straight 
switch-rail. 



Deg. of curve 
L 



762.75 
7° 31' 
84.75 



654.74 
8° 46' 
78.75 



733.68 
7° 49' 
75.68 



616.63 
9° 18' 
72.27 



This shows that the effect of using straight frog-rails and 
straight switch-rails is to sharpen the curve and shorten the lead 
in each case separately, and that the combined effect is still 
greater. The effect of the straight switch-rails is especially 
marked in reducing the length of lead, and therefore Eq. 78 to 
80, although having the advantage of extreme simplicity, can- 
not be used for point-switches without material error. The 
effect of the straight frog-rail is less, and since it can be mate- 
rially reduced by bending the free end of the frog-rails, the in- 
fluence of this feature is frequently ignored, the frog-rails are 
assumed to be curved, and Eq. Sb and 86 are used. (See § 276 
for a further discussion of this point.) 

267. Dimensions for a turnout from the outer side of a curved 




track. In this demonstration the switch-rails will be considered ii 
as uniformly circular from the switch-points to the frog-point. ' I 



§ 267. SWITCHES AND CROSSINGS. 303 

In the triangle FCD (Fig. 141) we have 
(FC-^CD) :(FC -CD) -tan ^(FDC+DFC) :tan HFDC-DFC); 
but i(FDC + DFC) = 90° - }^ 

and i(FDC-DFC)=iF. 

Also, FC + CD=2R and FC-CD=g; 

,-. 2R :g ::cot i^ : tan |i^ 
:: cot J-F : tan J^; 

.-. tanj^ = -g (91) 

Also, OF : FC :: sin l9 : sin ^; but ^ = (F-d); 

then r + j^ = (7^ + i^)_^5^ (92) 

5i^=L=2(i^ + }gr)sinJ^ (93) 

If the curvature of the main track is very sharp or the frog 
angle unusually small, F may be less than 6 ; in which case the 
center will be on the same side of the main track as C, Eq. 
92 will become (by calling r=—r and changing the signs) 

ir-ig)=(Ji+i9)^^~ (94) 

If we call d the degree of curve corresponding to the radius 
r, and D the degree of curve corresponding to the radius R, also 
d' the degree of curve of a turnout from a straight track (the frog 
angle F being the same), it may be shown that d=d^—D (very 
nearly). To illustrate we will take three cases, a number 6 
frog (very blunt), a number 9 frog (very commonly used), and a 
number 12 frog (unusually sharp). Suppose D=4° 0'; also 
Z) = 10° 0'; g=4' 8V'=4'.708. 

A brief study of the tabular form on p. 304 will show that the 
error involved in the use of the approximate rule for ordinary 
curves (4° or less) and for the usual frogs (about No. 9) is really 
insignificant, and that, even for sharper curves (10° or more), 
or for very blunt frogs, the error would never cause damage, 
considering the lower probable speed. In the most unfavorable 
case noted above the change in radius is about 1%. On account 
of the closeness of. the approximation the method is frequently 
used. The remarkable agreement of the computed values of L 



304 



RAILROAD CONSTEtrCTION. 



§267. 



Frog 




i) = 4°. 




"L'^ for 

straight 
track. 


num- 
ber. 


d 


d'-D 


Error. 


L 


6 

9 

12 


12° 54' 20" 
3 30 27 
13 33 


12° 57' 52" 
3 31 04 
13 36 


0° 03' 32" 
37 
03 


56.57 

84.85 

112.72 


56.50 

84.75 

113.00 



Frog 


D = 10°. 


"Z."for 
straight 
track. 


num- 
ber. 


d 


d'-D 


Error. 


L 


6 

9 

12 


6° 53' 24" 
2 27 54 
5 44 26 


6° 57' 52" 
2 28 56 
5 46 24 


0° 04' 28" 
01 02 
01 58 


56.66 

84.86 

112.91 


56.50 

84.75 

113.00 



with the corresponding values for a straight main track (the lead 
rails circular throughout) shows that the error is insignificant in 
using the more easily computed values. 

268. Dimensions for a turnout from the INNER side of a curved 
track. (Lead rails circular throughout.) From Fig. 142 we 
have, from the triangle DFC, 




Fig. 142. 

DC+FC: DC-FC :: tan i{DFC+FDC) : tan HDFC-FDC) ; 
but h(DFC + FDO = 90° - id 

and i(DFC-FDC)=iF; 

.\ 2R : g ::cot i<9 itsmiF 
iicotiF: tan id; 
gn 



tan J^ = 



R' 



(95) 



269. 



SWITCHES AND CROSSINGS. 



305 



From OFC, 



OF:FC::smQ:sm(F+d). 
sin 6 



(r + ig)={R-i9)-r. 



^sm{F-\-dy 
L=BF=2{R-ig)siii id. 



(96) 
(97) 



As in § 267, it may be readily shown that the degree of the 
turnout (d) is nearly the sum of the degree of the main track (D) 
and the degree (d^) of a turnout from a straight track when the 
frog angle is the same. The discrepancy in this case is some- 
what greater than in the other, especially when the curvature 
of the main track is sharp. If the frog angle is also large, the 
curvature of the turnout is excessively sharp. If the frog angle 
is very small, the liability to derailment is great. Turnouts to 
the inside of a curved track should therefore be avoided, unless 
the curvature of the main track is small. — — 

269. Double turnout from a straight track. In Fig. 143 the 
frogs Fi and Fr are generally made equal. Then, if there are 




Fig. 143. 

uniform curves from 5' to Fi and from B to Fr, the required 
value of Fjn iiS obtained from 



vers iF„ 



2{r-\-igy 



(98) 



r being found from Eq. 78, in which n is the frog number of Fi 
or Fr. 

MFm^^rtsLn^Fm; 

but since rtm =i cot iFm, 



MFm = 



2nn 



(99) 



306 RAILROAD CONSTRUCTION. § 269. 

Since vers i^i=^^:^, 

vers iFni = i vers Fi (100) 

Also, since (CiFmy = (MFmy + (CiM)\ we have 

Simplifying and substituting, r=2gn^, we have 



Dropping the J, which is always insignificant in comparison with 
2n^f we have 

nm==-^=nX.707(approx.) (101) 

Frogs are usually made with angles corresponding to integral 
values of n, or sometimes in ''half sizes, e.g. 6, 6 J, 7, 7}, etc. 
If No. 8J frogs are used for Fi and Fr^ the exact frog number 
for Fm is 6.01. This is so nearly 6 that a No. 6 frog may be used 
without sensible inaccuracy. Numbers 7 and 10 are a less 
perfect combination. If sharp frogs must be used, 8} and 12 
form a very good combination 

If it becomes necessary to use other frogs because the right 
combination is unobtainable, it may be done by compounding 
the curve at the middle frog. Fi and Fr should be greater 
than iFm- If equal to hFm, the rails would be straight from 
the middle frog to the outer frogs. In Fig. 144, 6i=Fi — iFm* 



Drawing the chord FiFm, 

KFjFm =Fi- hdj=Fi - iFi + iFm = i(Fi + ^Fm) ; 
KFm _ g 



^^^""sinOT;;"2sinKi^z + ^i^^)- ' ' ^^^'^ 
KFi^KFm cot KFiFm==i9 cot i(Fi + iFm); . (103) 

(r I In^ ^ ^^^^ I . 

KTi -t- ^^; 2 sin ^0 4 sin ^{Fi + Wm) sin \{Fi -iFm) 

^ .... (104) 



cos J-Pm— COsi^i 



§ 270. 



SWITCHES AND CROSSINGS. 



307 



If three frogs, all different, must be used, the largest may be 
selected as Fm ; the radius of the lead rails may be found by an 
inversion of Eq. 98; Fm niay be located in the center of the 
tracks by Eq. 99; then each of the smaller frogs may be located 




Fig. 144. 

by separate applications of Eq. 102 or 103, the radius being 
determined by Eq. 104. 

270. Two turnouts on the same side. In Fig. 145, let 0^ 
bisect O2D, Then (n + J^) = J(^2 + iS') ; also, Ofi^^O^Fi smd 
Fr^Fi. 

_ ^9 . 



YeTsFm = -r 



(105) 



BFm=(T'-\-\g)miF^ (106) 

It may readily be shown that the relative values of Fr, Fi, and 
Fm are almost identical with those given in § 269; as may be 




apparent when it is considered that the middle switch may be 
regarded simply as a curved main track, and that, as developed 



308 



RAILROAD CONSTRUCTION. 



§271. 



in § 267, the dimensions of turnouts are nearly the same whether 
the main track is straight or slightly curved. 

271. Connecting curve from a straight track. The "con- 
necting curve'' is the track 
lying between the frog and 
the side track where it be- 
comes parallel to the main 
track (FS in Fig. 146 or 147). 
Call d the distance between 
track centers. The angle 
FO^R=F (see Fig. 146). 
Call r' the radius of the con- 
necting curve. Then 




(r'-i^) = 



versi^' 



■ (107) 
FR = (r'- 



FiG. 146. 



■ig)smF. 



(108) 



If it is considered that the distance FR consumes too much 
track room it may be shortened b.y the method indicated in 
Fig. 151. 

272. Connecting curve from a curved track to the OUTSIDE. 
When the main track is curved, the required quantities are the 




radius r of the connecting curve from F to S, Fig. 147, and its 
length or central angle. In the triangle CSF 

CS + CF:CS'-CF::tsin i(CFS-\rCSF):tm HCFS-CSF); 



n 



§ 273. SWITCHES AND CROSSINGS. 309 

but i(CFS + CSF) =90 -i(p; and, since the triangle ^SF ia 
isosceles, i(CFS - CSF) = ^F; 

.-. 2R + d:d-g::cot i^:tan ^F 
::cot JF:tan i<p; 

••• *-i^='-S3^ <io«) 

From the triangle COiF we may derive 

r — ig:R-\-ig::sin ^:sin (i^+^); 

^-^^ = ^^ + i^)ih^W^- . o . . . (110) 

Also i^6^=2(r-i.g)sinKi^ + ^) (Ill) 

273. Connecting curve from a curved track to the INSIDE. 




Fig. 148. 

As above, it may readily be deduced from the triangle CFS (see 
Fig. 148) that 

(2R-d): (d -g) : : cot J^ : tan iF, 
and finally that 

*-*^-;?:^f ("2) 

Similarly we may derive (as in Eq. 110) 



(r-i9)HR-i9)^^^y . . 



(113) 



310 



RAILROAD CONSTRUCTION. 



§273. 



Also 



FS=2(r-ig)smi(F-(P) (114) 



Two other cases are possible, 
becomes infinite (see Fig. 149), 
then F = (lf, In such a case 
we may write, by substitut- 
ing in Eq. 112, 

2R-d=4.n\d-g). .(115) 

This equation shows the value 
of Rf which renders this case 
possible with the given values 
of n, df and g, (h) (p may be 
greater than F, As before 
(see Fig. 150) 

2R-d:d-g::cot J^itanji^; 
the same as Eq. 112, but 



(a) r may increase until it 




Fig. 149. 



r+igHR-ig)^ 



sin (ff 



sin ((fi -FY 



(116) 




Fig. 150. 

Problem, To find the dimensions of a connecting curve run- 
ning to the INSIDE of a curved main track; number 9 frog, 4° 30' 
curve, d = 13', 9r=4'8i''. 



§274. 



SWITCHES AND CROSSINGS. 



311 



Solution. 

Eq. 112. 



ff=13.000 

0=: ^ 4.708 

(d-g)= 8.292 



i2 = 1273.6 
2^ = 2547.2 
2R-d = 25S4.2 
log (222 -d) = 3. 40384 



Eq. 116. B = 1273.6 

i^= 2.35 

(R-y) = 1271.25 

(*_i?') = 1373". log = 3. 13767 

4.68557 

log sin (*-i?') = 7.82324 



Eq. 114. 



.2.83664 

4.68557 

=7.5222r 



log 2n = 

log (d-g) = 

co-log (2R-d) = 

log tan ^4^ = 



=1.25527 
= ,91866 
= 6.59616 

=8.77009 
3° 22' 14" 
6° 44' 28" 
6° 21' 35 " 
0° 22' 53' 



/og(i2-i^) = 3.10423 

log sin * = 9. 06960 

co-log sin (^-F) = 2.17676 

(r + ^g) = 22418.0. .4.35059 

r = 22415.6 

d = 0° 15' 

2... 0.30103 

(r+i£^) = 22418.0...4.35059 

sin i(<fr -F), .. 7.52221 

ir^=149 .22 2 . 17 384 



274, Crossover between 



9' 






V 








'^- 












IV 




Q 1 






\ 








X 


s 

/ 




// 

Y 


^N 








\ 


--d- 


i 


e^ 


S, 






D 






Oi 



Fig. 151. 



two parallel straight tracks. (See 
Fig. 151.) The turnouts 
are as usual. The cross- 
over track may be straight, 
as shown by the full lines, 
or it may be -a reversed 
curve, as shown by the 
dotted lines. The reversed 
curve shortens the total 
length of track required, 
but is somewhat objection- 
able. The first method re- 
quires that both frogs must 
be equal. The second 
method permits unequal 
frogs, although equal frogs 



are preferable. The length of straight crossover track is F^T, 



FiT sin Fi+g cos F^ =^d-g; 



^'^-^-^-*^'- 



(117) 



312 RAILROAD CONSTRUCTION. § 274. 

The total distance along the track may be derived as follows: 

XY = (d-g) cotF^; XF2=g^smF2; 

/. DV=2DF, + (d--g)cotF, — r-^ (118) 

sinF2 

If a reversed curve with equal frogs is used, we have 

vers^=|:; (119) 

also DQ==2rsm0 (120| 




Fig. 152. 

If the frogs are unequal, we will have (see Fig. 152) 
rj vers d-^-r^ vers d==d; 

d 



vers o — - 



also the distance along the track 

-S2^ = (n+^2) sin^. 



(121) 



(122) 



Problem. A crossover is to be placed between two parallel 
straight tracks, 12' 2" between centers, using a No. 8 and a No. 9 



§274. 



SWITCHES AND CROSSINGS. 



313 



frog, and with a reversed curve between the frogs. Required 
the total distance between switch-points (the distance B2N in 
Fig. 152). 

Solution. If straight point rails and straight frog rail^ are 
used, the radii, r^ and rj, taken from the middle section of Table 
IIL are 527.91 and 681.16. 



Eq. 122. 


vers ^== — -- 
ri + r2 


n = 527.91 


J=12'2" =12.16. log =1.08517 


rz = 681.16 


logrri+r2) = 3.08245 


rj+r2= 1209.07 


log vers ^ = 8.00272 






Eq. 122. 


log(ri+r2) =3.08245 




log sin <9 =9.15077 


^o,Y=l 


71.09 log 171.09 = 2.23322 



The length of the curve from B., = 100(<9 ~-d)= 100(8° 08' 06" 4- 
8° 250 =96.65. The length of the other curve is 100(8° 08' 06' 4- 




FiG. 153. 



10° 52') =74.86. As a check, 96.65 + 74.86 = 171.51, which is 
slightly in excess of 171.09, as it should be. 



314 



RAILROAD CONSTRUCTION. 



§275. 



275. Crossover between two parallel curved tracks, (a) Using 
a straight connecting curve. This solution has limitations. If 
one frog (F^) is chosen, F2 becomes determined, being a function 
of Fi. If Fi is less than some limit, depending on the width (d) 
between the parallel tracks, this solution becomes impossible. 
In Fig. 153 assume F^ as known. Then FyH=g sec F^. In the 
triangle HOF2 we have 

sin HF2O : sin F^HO r.HO'.F^O; 
sinF^HO^cosF,; HF^O =90'' ^F^; 
.". smHF20=cos F2. 
HO=R + id-ig-g sec F,; F20=R-id-hig; 
R + id -ig-g sec F, 



cos i^2==cos F^- 



R-id + ig 



(123) 



Knowing F2, 62 is determinable from Eq. 91. Fig. 153 shows 
the case where 62 is greater than F2. Fig. 154 shows the case 
where it is less. The demonstration of Eq. 123 is applicable to 




I Fig. 154. 

both figures. The relative position of the frogs F^ and F2 may 
be determined as follows, the solution being applicable to both 
Figs. 153 and 154: 

HOi^2 = 180°-(90°-Fi) -(90° + i^2) =^i--?^2. 
Then 

GF,=2{R + ld-lg)sml{F^-F2) (124) 

Since F2 comes out any angle, its value will not be in general 
that of an even frog number, and it will therefore need to be 
made to order. 



§275. 



SWITCHES AND CROSSINGS. 



315 



(b) Continuing the switch-rail curves until they meet as a 
reversed curve. In this case Fi and F2 may be chosen at pleasure 
(within hmitations), and they ^dll of course be of regular sizes 
and equal or unequal as desired. F^ and F2 being known, 6^ 
and 62 are computed by Eq. 95 and 91. In the triangle 00 fi^ 
(see Fig. 155) 

, 2{S-002){S-00,) 
vers^- {^002){00,) ' 

in which S = ^{00^ + 00^ + 0,02) ) 

but 00i==E + ic^-ri, 

002=ie-J^+r2, 

.-. >S: = K2i^+2r2)=i^+r2; 
S-002=n + r2-R-\-hd-r2 = hd) 
S-00^=R-{-r2-R-id-hr^=ri-hr2-id; 




vers (p = 



Fig. 155. 



(i^-Jd + r2)(E + id-ri)' 



,00i 



=sm (p i; 



Ofi.D^if+Ofifi; 



(125) 

(126) 

(127) 
(128) 



316 



RAILROAD CONSTRUCTION. 



§275. 



Although the above method introduces a reversed curve, yet 
it uses up less track than the first method and permits the use of 
ordinary frogs rather than those having some special angle which 
must be made to order. But the above solution implies the use 
of circular lead rails. We may compute dimensions and lay 
track between F^ and F2 on this basis and then change the 
switch rails as desired. Strictly, r^ and rg should be computed 
by Eq. 92 and 96, but for an easy main-line curve the approxi- 
mate rule is sufficiently accurate. 

Problem. — Required the dimensions of a crossover on a 
4° 30' curve when the distance between track centers is 13 feet. 
The frog for the outer main track (F^ in Fig. 155) is No. 9; 
F2 is No. 7. Then E= 1273.6; R^, for the outer main track, 
= 1280.1; Di = 4°29'; 7^2=1267.1; D2=4°3V; r^= radius for 
(d,-\-D,y curve=radius for (7° 31^ + 4° 290 curve=478.34; 
r2=radius for {d^-D^Y curve=radius for (12° 26' -4° 31? 
curve=724.31. (See §§ 267-268.) 



Eq. 125 

ri + r2-ic; = 1196.15 
iB-irf+r2 = 1991.31 
R + \d-n= 801.76 

4>=^7° 52' 2^" 

Eq. 126 
ri + r2 = 1202.65 



d = 13 

log = 3. 30914 
log = 2.9040i 



log = 3. 08013 
0020i = 5° \M 2^' 



log = l. 11394 

log = 3. 07778 

colog = 6.69086 

colog = 7 09595 

l og vers = 7-97854 

log sin = 9-13670 

log(i2 + tc?-ri) = 2.90404 

colog = 6-91986 

sin 0020i = 8. 96061 



Eq. 127 O2O1D =7° 52' 26" + 5° 14' 24" = 13° 06' 50" 



Eq. 91 



^^ = 3047' 30" 



tan i $1 ■■ 



gn 
~ R 



4.708X9 ^ 42.372 log = l. 62708 

1280.1 1280.1 log = 3. 10724 

log tan i ^1 = 8. 51983 

5.31426 

log 6825 = 3-83410 



(Using Table VI) 
Hi = l°53'45" 



Eq. 95 



^2 = 2° 58' 48" 



gn 
B 



4.708X7 
1267.1 



32.956 
^1267.1 



Eq. 128 



JVF2 = 48.84 



i^2 = l°29'24" 

i2-i^ + i^ = 1269.45 
= 1° 06' 08" 



log = l. 51795 

log = 300281 

tan +^2 = 8-41512 

5-31433 

log 5364 = 3. 72945 



2 log = 0. 30103 

log = 3. 10361 

1 . 4.68555 
logsin = 

3-59857 

V>g 48. 84 = 1.68876 



§ 276. SWITCHES AND CROSSINGS. 317 

13° 06' 50'' 
Length of curve with radius rj = 100 ^ ^ — = 109 . 18 

•• " " " •• r2-100 ^ ^^^f —= 66.16 



Total length of curve between switch points = 175.34 

As an approximate check, the mean length subtending the 
angle ip with radius R is similarly computed as 174.98. Note 
that the length of the curve T\ith the radius r^ is 66.16, which 
is but little more than the length of lead rails (65.92) for a No. 7 
frog using circular lead rails, which means that the point of 
reversed curve is but little beyond the frog point. If the 
computations had apparently indicated the point of reversed 
curve coming be ween the frog point and the switch point, it 
would have shown the impracticability of the combination 
of No. 7 and No. 9 frogs with this particular degree of curve 
gauge of track, and distance between track centers. If both 
frogs were made No. 9, the total length of track between switch 
points would be increased to over 198 feet and the point of 
reversed curve would be nearly at the middle point. This 
shows that the frog numbers should be nearly equal, but also 
shows that there is some choice '^within limitations." 

276. Practical rules for switch-laying. A consideration of 
the previous sections will show" that the formulae are compara- 
tively simple when the lead rails are assumed as circular; that 
they become complicated, even for turnouts from a straight 
main track, when the effect of straight frog and point rails is 
allowed for, and that they become hopelessly complicated when 
allowing for this effect on turnouts from a curved main track. 
It is also shown (§ 267) that the length of the lead is practically 
the same whether the main track is straight or is curved with 
such curves as are commonly used, and that the degree of curve 
of the lead rails from a curved main track may be found with 
close approximation by mere addition or subtraction. From 
this it may be assumed that if the length of lead (L) and the 
radius of the lead rails (r) are computed from Eq. 87 and 90 for 
various frog angles, the same leads may be used for curved main 
track; also, that the degree of curve of the lead rails may be 
found by addition or subtraction, as indicated in § 267, and that 
the approximations involved will not be of practical detriment. 
In accordance with this plan Table III has been computed from 



318 



RAILROAD CONSTRUCTION. 



§ 276. 



MN=fc 

yMDN=(X 
VHIVIR=M(F-a) 




Eq. 87, 88, and 90. The leads there given may be used for all 
main tracks, straight or curved. The table gives the degree of 

curve of the lead rails for straight 
main track; for a turnout to the 
inside, add the degree of curve of the 
main track; for a turnout to the out- 
side , subtract it. 

But there are complications result- 
ing from practical and economical 
switch construction. A committee of 
of the A. R. E. A. in 1910 adopted 
certain standards in details, which 
when appHed to Eqs. 87 to 90 give 
the values for switch dimensions as 
quoted in the second section of Table 
III. They adopted four lengths of 
switch rails. In each case the '* point '' 
Fig. 140. is always J" thick. The gauge hne 

at the other end is always to be placed 6i" from the gauge line 
of the main rail, and the planing is so done that when in this 
position the switch rail lies against the main rail. Therefore 
the angle a is alwaj^s an angle whose sine equals 6 inches (or 
0.5 foot) divided by the length of the switch rail in feet. In 
Fig. 140, the point D is not on the gauge line of the main rail 
but at a point J" away from it, and the point M GJ" away from 
it. The straight rail BF consists of a point rail 
at one end, the '^closure rails,'' and one of the wing 
rails of the frog at the other end. The closure rails 
will in general consist of one rail cut to a computed 
length and one or more rails from 24 to 33 feet long, 
the lengths being in even feet. The curved rail DF 
will also consist of a point rail, a frog wing-rail, and 
one or more lengths of closure rail, but the closure 
rails in this case are slightly longer than those for the 
straight rail. Since it is always practically easier 
to measure to the '^ actual point'' of a frog (see 
Fig. 130) rather than to the theoretical point, 
Table III gives the distance L', which is the dis- 
tance L, =BF, plus the ^^frog bluntness," which 
is found by multiplying I" ( = 0.0417 foot) by the frog number. 
The curvature for a curved switch rail (for a straight track) 




Fig. 156. 



§ 276. 



SWITCHES AND CROSSINGS. 



319 



SWITCH EAIL. 



Frog 
No. 


Center. 


Quarter 
points. 


Ft. 

.59 
.55 
.50 
.58 
.55 
.50 
.49 
.48 
.55 
.52 
.61 
.53 
.50 
.44 


Ins. 

P 

7 

61 

6 

51 

5f 

6- 

6i 

7- 

6- 

I 


Ft. 

.44 
.41 
.38 
.43 
.41 
.38 
.37 
.36 
.41 
.39 
.46 
.40 
.37 
.33 


Ins. 

5f 

5 

4^ 

1' 

4i 
4- 
4- 
4- 
4f 
5- 
4- 
4* 
4 


4 
5 
6 
7 

8 

9 

91 
10 
11 
12 
15 
18 
20 
24 



ORDiNATES OF CURVED may be determined by measuring off- 
sets at the quarter points from a string 
stretched between the points FI and M. 
These offsets are as given in the tabular 
form. The distance HM may be com- 
puted from the equation in § 265. 

If the position of the switch-block is 
definitely determined, then the rails 
must be cut accordingly; but when 
some freedom is allowable (which never 
need exceed 16.5 feet and may require 
but a few inches), one rail-cutting may 
be avoided. Mark on the rails at B, 
F, and D; measure off the length of 
the switch-rails DN and locate the point 
M at the distance k from N. If the frog must be placed 
during the brief period between the running times of trains, it 
will be easier to joint up to the frog a piece of rail at one or 
both ends of just such a length that they may be quickly substi- 
tuted for an equal length of rail taken out of the track. When the 
frog is thus in place the point H becomes located. The chord 
MH may be measured on the ground as a check. The curve 
between M and H is of known radius. Substituting in Eq. 31 
the value of chord and Ry we may compute x{ = dh). Locate the 
middle point d and the quarter points a" and c". Then a^'a 
and c''c each equal three-fourths of dh. Theoretically this 
gives a parabola rather than a circle, but the difference for all 
practical cases is too small for measurement. See Fig. 156. 

Example. — Given a main track on a 4° curve — a turnout to 
the outside, using a No. 9 frog; gauge 4' 8^"; / = 6'.00; /c = 6i"; 
DN = W 6" and a = V 44' 11". Then for a straight track r 
would equal 616.27 [d = 9° 18' 27"]. For this curved track d 
will be nearly 9° 18'-4° = 5° 18', or r will be 1081.4. L for the 
straight track would be 72.61; but since the lead is slightly in- 
creased (see §267) when the turnout is on the outside, of a curve, 
L may here be called 72.9. H and M may be located as de- 
scribed above. HM may be measured on the ground as a check 
on the value (in this case 49.90) computed as in § 265. Since 
it is sKghtly more for a turnout to the outside of a curve, it may 

(50.0)^ 
be called 50.00. Then, for the outer rail, x = dh= Q\yiAQo yg = 

0.288 feet, and aa'' and cc" =0.216 foot. 



320 



EAILROAD CONSTRXJCTION. 



§ 277 



276a. Slips. Track movements in crowded yards are facili- 
tated by using ^^slips" (see Fig. 156a;, which may be ^'single'' 
or '' double." The crossing of two rails is done either by 
operating two movable rails or by using fixed ''frogs/' but a 
comparison of the continuity of the running rails, using 
ordinary frogs (see Fig. 130) and these frogs, will show their 
radical difference. These slips can be used for frog angles from 
No. 6 to No. 15. The levers are so connected that the several 
operations necessary to set the rails for any desired train move- 
ment are accomplished by one motion. 



CROSSINGS. 

277. Two straight tracks. When two 

straight tracks cross each other, four frogs 
are necessary, the angles of two of them 
being supplementary to the angles of the 
other. Since such crossings are sometimes 
operated at high speeds, they should be 




276a. 



SWITCHES AND CROSSINGS. 



321 




FiQ. 156a. — Single and Double Slips. 



322 



KAILROAD CONSTRUCTION. 



§ 279. 



Structurally the 



very strongly constructed, and the angles should preferably be 
90° or as near that as possible. The frogs will not in general 
be ^' stock'' frogs of an even number, especially if the angles are 
large, but must be made to order with the required angles as 
measured. In Fig. 157 are shown the details of such a crossing. 
Note the fillers, bolts, and guard-rails. 

278. One straight and one curved track, 
crossing is about the same as above, 
but the frog angles are all unequal. 
In Fig. 158, R is known, and the 
angle ikT, made by the center lines 
of the tracks at their point of inter- 
section, is also known. 

M=NCM, NC=R cos M, 

(R-ig) cos F,=NC + ig; 

R cos M + ig 



,\ cosFi = 
Similarly 

cos F2 = 

cos Fo = 



cos F^ = 



R-ig ' 

R cos M + jg 

R + ig ' 

R cosM—jg 

R+hg ' 

R cos M — ^g 
R-ig . 



(129) 




F,F, = (R + ig) sin F,-(R-ig) sinF,; \ 



(130) 



Hi^, = (i^-i^)(sini^,-sini^i). f • 

279. Two curved tracks. The four frogs are unequal, and 
the angle of each must be computed. The radii Ri and R2 are 
known; also the angle M. rj, rg, rg, and r^ are therefore known 
by adding or subtracting ^(7, but the lines are so indicated for 
brevity. Call the angle ikfCjCz = (7i, the angle MC2Ci = C2, and 



the line CiC2=c. 



and 



Then 
i((7i + (72)=90°-iilf 

tan i(Ci-C2) =cot ^M^ 



-Ri 



R2-\-Ri 



Ci and C2 then become known and 
C = C iC 2 ^= -n/2~ 



''sin C 



(131) 



(132) 



279. SWITCHES AND CROSSINGS. 323 

In the triangle FiC^Ci, call Kc+n + rJ =5i; S2 = i(c + r2+r^\ 







V K / ^ 

V !| / "\ \\ \ 



\ ll 

\ ri / 
\ h / 



S3 = f:(c4-ri+r3) 
Table XXX, 

Similarly 



Fig. 159. 
; and s^ = i(c-\-r2+r2). Then, by formula 29, 

yersi^,= '^^--^-^^^^-^-\ ^ 

versi^,^ '^^--^-^^^--^-\ 

vers F^ = ^^ ^-^^-^ -, 



vers i^4 = 
sin 



2(g,-r.)(6 -, 



-r^) 



(7iC2i^,=sini^,^' 



. (133) 






sin (7iC2F2=sin i^2~; 

/. F^C^F^^CS^^F^-CfiJ^.,, (134) 

sin i<'iCi(72=sin jPj— ; 

sini^2C'A=sini^2-, 

.-. F,C,F2=F,C,C,-F,Cfi2) ' .... (135) 
from which the chords F^F2 and i^2^4 are readily computed. 



324 



RAILROAD CONSTRUCTION, 



§ 279. 



F1F2 and 7^2^4 ^^^ nearly equal. When the tracks are straight 
and the gauges equal, the quadrilateral is equilateral. 

Problem. Required the frog angles and dimensions for a cross- 
ing of two curves (Z)i=4°; 2)2 = 3°) when the angle of their tan- 
gents at the point of intersection =62*^ 28' (the angle M in 
Fig. 159). 

Solution 

i^i = 1432.7; 2^2 = 1910.1; 
n =-^2 + 1^ = 1910. 1+2.35 = 1912.45; 
^2 =-R,-i^ = 1910. 1-2.35 = 1907.75; 
rg =J?i + i^ = 1432. 7 + 2. 35 = 1435. 05; 
r^ =J?i-i^ = 1432. 7-2. 35 = 1430. 35. 

Eq. 131. log cot JM=0. 21723 

Eo-T^i =477.4; log =2.67888 

ie2 + i?i =3342.8; log=3. 52411; co-log = 6 . 47589 
KC'1-^2) =13° 15' or'; tan 13° 15' 07" = 9. 37200 
h{C.-\-C,) =58° 46' [KC'i + C2) =90°-JM] 

Ci=72°01'07" 
(72=45° 30' 53" 



Eq. 132. 



log sin Ci=9 



c = C,C2 = 1780.7; 

Eq. 133. 
c=17S0.7 
ri = 1912.45 
r4 = 1 430.35 
_2|5123.50 
si = 256 1.75 
si-ri= 649.30 



logi?2=3.28105 

log sin ilf= 9. 94779 

.97825; co-log = 0.02175 

logCiC2=3.25059 



si-r4= 1131.40 



c = 1780.7 
r2= 1907.75 
r4= 1430.35 
_2|5118.80 
52 = 2559.40 
r2= 651.65 
-r4= 1129.05 



S2- 



ri = 1912.45; log = 3. 28159; 
n - 1430 . 35 ; log = 3 . 15544 ; 
F, ^6 2° 25^ ?A" 



c=1780.7 

r2= 1907.75 

r3 = 1435.05 

_2|5123.50 

§4 = 2561. 75 

S4~r2= 654.00 

S4-r3-n26.70 

log 2 = 0.30103 

(si-ri); log 649.30 = 2.81244 

(si-r4); log 1131.40 = 3.05361 

co-log = 6. 71841 

co-log = 6.S4456 

log vers 62° 25 ^ 31^^ = 9. 73006 



c = 


= 1780 


7 


ri = 


= 1912 


45 


r3 = 


= 1435 


05 


2|5128 


20 


53 = 


= 2564 


10 


r,= 


= 651 


65 


r3 = 


= 1129 


05 





log 2 = 0.30103 




(s2-r2); log 651.65 = 2.81401 




(s2-r4); log 1129.05 = 3.05271 


r2= 1907.75; log = 3. 28052; 


co-log = 6. 71948 


r4= 1430.35; log = 3. 15544] 


co-log = 6. 84456 


F<. = 62^ 33' 55"; 


log vers 62° 33' 55" = 9. 73180 



§ 279. RAILROAD CONSTRUCTION 325 

log 2 = 0.30103 

(s3-n); log 651.65 = 2.81401 

(ss-rs); log 1129.05 = 3.05271 

ri = 1912.45; log = 3. 28159; co-log = 6. 71841 

r3= 1435.05; log = 3. 15686; co-log = 6_84313 

F3 = 62° ^2r Sr ;^; • log vers 62° 21 ^ 57^^ = 9.72930 

log 2 = 0.30103 

(sA-r-z); log 654.00 = 2.81558 

(«4-r3); log 1126.70 = 3.05181 

r2=1907.75; log = 3. 28052; co-log = 6.71948 

rs = 1435 . 05 ; log = 3 . 15686 ; co-log = 6.84313 

F4 = 62° 30^ 14^^ log vers 62° 3 0^ 14^^ = 9.73103 

As a check, the mean of the frog angles = 62° 27' 54 ', which is within 6" of 
the value of M. 

Eq. 134. log sin i?4 = 9. 94794 

log r3 = 3. 15686 

log c = 3 . 25059 ; co-log c = 6 . 74940 

CiC2^4 = 45° 37' 51"; sin 0102^4 = 9.85421 



log sini^2 = 9.94818 

log r4 = 3. 15544 

co-log c = 6. 74940 

CiC2F2 = 45° 28' 17"; sin C1C2F2 = 9 . 85303 

^2(72^^4 = 45° 37' 51" -45* 28' 17" = 0° 09' 34" . 

log 2 = 0.30103 
log r2 = 3. 28052 

i(0° 09' 34" ) = 0° 04' 47" ; log sin = (^^ • 5?5il 

\^ .4o/8o 

^2^^4 = 5. 309 ; log F2F4 = 0.72500 

Eq. 135. sin Fi = 9. 94763 

log ri = 3. 281 59 

co-log c = 6. 74940 

FiCiC72=72° 10' 22"3 sin i^iCi Co = 9^97863 

sin ^2 = 9. 94818 

log r2 = 3. 280.52 

co-log c = 6. 74940 

Fj,CiC2=71° 57' 38"; sin -F2CiC2 = 9.97811 

FiCiF2 = 72° 10' 22" -71° 57' 38" = 0° 12' 44" . 

log 2 = 0.30103 
log r4 = 3. 15544 

K0° 12' 44") = 0° 06' 22"; log pin= Z^. 68557 

V 2.58206 
FiFi = 5.298; logFiF2 = 0.72411 

As a check, i^2^4 ^^^ ^i^z ^re very nearly equal, as they should 
be. 



CHAPTER XII. 

MISCELLANEOUS STRUCTURES AND BUILDINGS. 
--> WATER-STATIONS AND WATER-SUPPLY. 

280. Location. The water-tank on the tender of a locomo- 
tive has a capacity of from 3000 to 7000 gallons — sometimes less, 
rarely very much more. The consumption of water is very vari- 
able, and will correspond very closely with the work done by 
the engine. On a long down grade it is very small; on a ruling 
grade going up it may amount to 150 gallons per mile in ex- 
ceptional cases, although 60 to 100 gallons would be a more usual 
figure. A passenger locomotive can run 60 miles or more 
on one tankful, but freight work requires a shorter interval 
between water-stations. On roads of the smallest traffic, 
15 to 20 miles should be the maximum interval between stations J 
10 miles is a more common interval on heavy traffic-roads. But 
these intervals are varied according to circumstances. In the 
early history of some of the Pacific railroads it was necessary to 
attach one or more tank-cars to each train in order to maintain 
the supply for the engine over stretches of 100 miles and over 
where there was no water. Since then water-stations have been 
obtained at great expense by boring artesian wells. The indi- 1 
vidual locations depend largely on the facility with which a suffi- ' 
cient supply of suitable water may be obtained. Streams inter- 
secting the railroad are sometimes utilized, but if such a stream 
passes through a limestone region the water is apt to be too hard 
for use in the boilers. More frequently wells are dug or bored. 
When the local supply at some determined point is unsuitable, 
and yet it is necessary to locate a water-station there, it may 
be found justifiable to pipe the water several miles. The con- 
struction of municipal water-works at suitable places along the 
line has led to the frequent utilization of such supplies. In such 
cases the railroad is generally the largest single consumer and 
obtains the most favorable rates. When possible, water-stations 
aie located at regular stopping points and at division termini. 

326 



§ 281. MISCELLANEOUS STRUCTUKES AND BUILDINGS. 327 

281. Required qualities of water. Chemically pure water is 
unknown except as a laboratory product. The water supplied 
by wells, springs, etc., is always more or less charged with cal- 
cium and magnesium carbonates and sulphates, as well as other 
impurities. The evaporation of water in a boiler precipitates 
these impurities to the lower surfaces of the boiler, where they 
sometimes become incrusted and are difficult to remove. The 
protection of the iron or steel of a boiler from the fierce heat of 
the fire depends on the presence of water on the other side of the 
surface, w^hich w^ill absorb the heat and prevent the metal from 
assuming an excessively high temperature. If the water side 
of the metal becomes covered or incrusted vdih a deposit 
of chemicals, the conduction of heat to the water is much less 
free, the metal will become more heated and its deterioration or 
destruction will be much more rapid. An especially common 
effect is the production of leaks around the joints between tubes 
and tube-sheets and the joints in the boiler-plates. Such in- 
jury can only be prevented by the application of one (or both) 
of two general methods — (a) the frequent cleaning of the boilers 
and (b) the chemical purification of the water before its intro- 
duction into the boiler. Although ^^ manholes'^ and ^^hand- 
holes'' are made in boilers, it is physically impossible to clean 
out every corner of the inside of a boiler where deposits will form 
and where they are especially objectionable — on the tube-sheets. 
Such a cleaning is troublesome and expensive. 

Chemical purification is generally accomplished by treating 
the water before it enters the boiler. The reagents chiefly em- 
ployed are quicklime and sodium carbonate. Lime precipi- 
tates the bicarbonate of lime and magnesia. Sodium carbo- 
nate gives, by double decomposition in the presence of sulphate 
of lime, carbonate of lime, which precipitates, and soluble sul- 
phate of soda, which is non-incrustant. \ATien this is done in a 
purifying tank, the purified water is drawn off from the top of 
the tank and supplied pure to the engines. The precipitants are 
drawn off from the settling-basin at the bottom of the tank. 
This purification, w^hich makes no pretense of being chemically 
perfect, may be accomplished for a few cents per 1000 gallons. 
It is used much more extensively in Europe than in this country, 
the Southern Pacific being the only railroad which has employed 
such methods on a large scale. Reliance is frequently placed 
on the employment of a ^'non-incrustant'' which is introduced 



328 



RAILROAD CONSTRUCTION. 



§281. 



directly into the boiler. When no incrustation takes place 
the accumulation of precipitant and mud in the bottom of the 
boiler may be largely removed by mere ^^ blowing off'' or by 
washing out with a hose. 

On the other hand, there is the exceptional case that the 
water may be too pure. It is well known that distilled water 
has a very strong corrosive action on iron and that it is possible 
for the water to be so pure that corrosion of the boiler tubes 
will be accelerated and that the boilers will rapidly deteriorate 
in this way. It is therefore occasionally necessary to add a 
small portion of lime to a very isux water, so that a very thin 
scale will form over the surface of the iron, which will protect 
the iron from corrosion. 

American practice may therefore be summarized as follows: 
(a) Employing as pure water as possible; (b) cleaning out boil- 
ers by ^^ blowing off'' or by washing out with a hose or by physi- 
cal scraping at more or less frequent intervals or when other 
repairs are being made; (c) the occasional employment of non- 
incrustants; (d) the occasional chemical treatment of water 
before it enters the tender-tank. 

282. Tanks. Whatever the source, the water must be led 
or pumped into tanks which are supported on frames so that the 

bottoms of the tanks are 
about 12 feet above the 
rails. Wooden tanks hav- 
ing a diameter of 24 feet, 
16 feet high, and with a 
capacity of over 50,000 
gallons, are frequently 
employed. Iron or steel 
tanks are also used. 

In Table XIV is shown 
the capacity of cylindrical 
water-tanks in United 
States standard gallons of 
231 cubic inches. From 
this table the dimen- 
sions of a tank of any 
Fig. 160.— Water-tank. j • j -x 

desired capacity may 

readily be found. Two or more tanks are sometimes used 
rather than construct one of excessive size. The smaller sizes 




'- ^J, , ,..:Lr J 



§ 282. MISCELLANEOUS STKUCTURES AND BUILDINGS. 329 

TABLE XIV. CAPACITY OF CYLINDRICAL WATER-TANKS IN 

UNITED STATES STANDARD GALLONS OF 231 CUBIC INCHES. 



Height 






Diameter of tank in feet. 






m 
feet. 


10 


12 


14 


16 


18 


20 


22 


24 


6 


3525 


5076 


6909 


9024 


11421 


14101 


17062 


20305 


7 


4113 


5922 


8061 


10528 


13325 


16451 


19905 


23689 


8 


4700 


6768 


9212 


12032 


15229 


18801 


22749 


27073 


9 


5288 


7614 


10364 


13536 


17132 


21151 


25592 


30457 


10 


5875 


8460 


11515 


15041 


19036 


23501 


28436 


33841 


11 


6463 


9306 


12667 


16545 


20939 


25851 


31280 


37225 


12 


7050 


10152 


13819 


18049 


22843 


28201 


34123 


40609 


13 


7638 


10998 


14970 


19553 


24746 


30551 


36967 


43994 


14 


8225 


11844 


16122 


21057 


26650 


32901 


39810 


47378 


15 


8813 


12690 


17273 


22561 


28554 


35251 


42654 


50762 


16 


9400 


13536 


18425 


24065 


30457 


37601 


45498 


54146 


17 


9988 


14383 


19576 


25569 


32361 


39951 


48341 


57530 


18 


10575 


15229 


20728 


27073 


34264 


42301 


51185 


60914 


19 


11163 


16075 


21879 


28577 


36168 


44652 


54028 


64298 


20 


11750 


16921 


23031 


30081 


38071 


47002 


56872 


67682 


21 


12338 


17767 


24182 


31585 


39975 


49352 


59716 


71067 


22 


12925 


18613 


25334 


33089 


41879 


51702 


62559 


74451 


23 


13513 


19459 


26485 


34593 


43782 


54052 


65403 


77835 


24 


14101 


20305 


27637 


36097 


45686 


56402 


68246 


81219 


25 


14688 


21151 


28789 


37601 


47589 


58752 


71090 


84603 



shown in the table are of course too small for ordinary use, 
but that part of the table was filled out for its possible con- 
venience otherwise. On single-track roads where all engines 
use one track the tank may be placed 8' 5" from the track 
center; this gives sufficient clearance and yet permits the use 
of a single swinging pipe which will reach from the bottom 
of the tank to the tender manhole. In ]7ig» 160 is illustrated 
one form of wooden tank. They are preferably manufactured 
by those who make a special business of it and who by the use 
of special machinery can insure tight joints. When it is incon- 
venient to place the tank near the track, or when there is a 
double track, a '' stand-pipe '^ becomes necessary. See §285. 
One of the most difficult and troublesome problems is to prevent 
freezing, particularly in the valves and pipes Not only are the 
pipes carefully covered but fires must be maintained during cold 
weather. When the pumping is accomplished by means of a 
steam-pump, suppHed from a steam-boiler in the pump -house 
under the tank, coils of steam-pipe may be employed to heat the 
water or to heat the pipes Partial protection may be obtained 
by means of a double roof and double bottom, the spaces being 
filled with sawdust or some other non-conductor of heat. 



330 RAILROAD CONSTRUCTION. § 283. 

283. Pumping. The pumping is done most reliably with * 
steam-pumps or gas-engines, although hot-air engines,^windmills, 
and even man-power are occasionally employed. Economy of ; 
operation requires that the water-stations shall be so located 
that each tank shall be used regularly and that each pump shall 
be regularly operated for maintaining the water-supply. On ; 
the other hand, the pump should not be required to regularly | 
work at night to maintain the supply and should have an excess { 
capacity of say 25%. When a tank is but little used, it will still 
require the labor of an attendant, and his time will be largely 
wasted unless he can be utilized for other labor about the station. 
In recent years gasoline has been extensively emplo^^ed as a fuel 
for the pumping-engines. The chief advantages of its use lies in 
the extreme simplicity of the mechanism and the very slight 
attention it requires, which permits their being operated by 
station-agents and others, who are paid $10 per month extra, 
instead of paying a regular pumper $35 per month. " Screen- 
ings," "slack coal," etc., are used as fuel for steam-pumps and 
may frequently be deUvered at the pump-house at a cost not 
exceeding 30 cents per ton, but even at that price the cost of 
pumping per thousand gallons, although dependent on the hori- 
zontal and vertical distances to the source of supply and to 
the tank, will generally run at 2 cents to 6 cents per 1000 gallons. 
In many cases where steam plants have been replaced by gasoline 
plants, the cost of pumping per 1000 gallons has been reduced 
to one third or even one fourth of the cost of steam pumping. 
Of course the cost, using windmills, is reduced to the mere 
maintenance of the machinery, but the unreliability of wind as 
a motive power and the possibility of its failure to supply water 
when it is imperatively needed has made this form of motive 
power unpopular. (See report to Ninth Annual Convention 
of the Association of Railway Superintendents of Bridges and 
Buildings, Oct. 1899.) 

284. Track tanks. These are chiefly required as one of the 
means of avoiding delays during fast-train service. A trough, 
made of steel plate, is placed between the rails on a stretch of 
perfectly level track. A scoop on the end of a pipe is lowered 
from under the tender into the tank while the train is in motion. 
The rapid motion scoops up the water, which then flows mto the 
tender tank. The following brief description of an 'nstallation 
on the Baltimore & Ohio Railroad between Baltimore and 



§286. MISCELLANEOUS STRUCTURES AND BUILDINGS. 331 

Philadelphia will answer as a general description of the 
method. The trough is made of yV' steel plate, 19'' wide, 6" 
deep, and has a, length of 1200 feet. There is riveted on each 
side a line of lJ''X2''Xi''' angle bars. These angle bars rest on 
the ties. Ordinary track spikes hold these angle bars to the 
ties, but permit expansion as with rails. The tanks are firmly 
anchored at the center, the ends being free to expand or con- 
tract. The plates are 15 feet long and are riveted with jV 
rivets, 20 rivets per joint. At each end is an inclined plane 
13' 8" long. If the fireman should neglect to raise the scoop 
before the end of the tank is reached, the inclined plane will 
raise it automatically and a catch will hold it raised. Water 
is supplied to the tanks by a No. 9 Blake pump having a 
capacity of 260 gallons per minute. During cold weather, 
freezing is prevented by injecting into the side of the tanks, 
at intervals of 45 feet, jets of steam, which come through 
J" holes. Two boilers of 80 and 95 H.P. are required for pump- 
ing and to keep the water from freezing. During w^arm 
weather an upright 25 H.P. boiler suffices for the pumping. 
The cost of installation was about $10,000 to $11,000, the cost of 
maintenance being about $132.50 per month. 

285. Stand-pipes. These are usually manufactured by those 
who make a specialty of such track accessories, and who can 
ordinarily be trusted to furnish a correctly designed article. In 
Fig. 161 is sho^vn a form manufactured by the Sheffield Car Co. 
Attention is called to the position of the valve and to the device 
for holding the arm parallel to the track when not in use so that 
it will not be struck by a passing train. When a stand pipe is 
located between parallel tracks, the strict requirements of clear- 
ance demand that the tracks shall be bowled outward slightly. 
If the tracks were originally straight, they may be shoved over by 
the trackmen, the shifting gradually running out at about 100 
feet each side of the stand-pipe. If the tracks were originally 
curved, a slight change in radius will suffice to give the necessary 
extra distance betw^een the tracks. 

BUILDINGS. 

286. Station platforms. These are most commonly made of 
planks at minor stations. Concrete is used in better-class work, 
also paving brick. An estimate of the cost of a platform of paving 
brick laid at Topeka, Kan., was $4.89 per 100 square feet when 



332 



RAILROAD CONSTRUCTION. 



§286. 



laid flat and $7.24 per 100 square feet when laid on edge. The 
curbmg cost 36 cents per Hnear foot. Cinders, curbed by timbers 




Fig. 161. — Stand-pipe. 
or stone, bound by iron rods, make a cheap and fairly durable 
platform, but in wet weather the cinders will be tracked into 



§ 287. MISCELLANEOUS STRUCTURES AND BUILDINGS. 333 

the stations and cars. Three inches of crushed stone on a 
cinder foundation is considered to be still better, after it is once 
thoroughly packed, than a cinder surface. 

Elevation. — The elevation of the platform with respect to 
the rail has long been a fruitful source of discussion. Some roads 
make the platforms on a level with the top of the rail, others 
3'' above, others still higher. As a matter of convenience to 
the passengers, the majority find it easier to enter the car from 
a high platform, but experience proves that accidents are more 
numerous with the higher platforms, unless steps are discarded 
altogether and the cars are entered from level platforms, as is 
done on elevated roads. As a railroad must generally pay 
damages to the stumbUng passenger, they prefer to build the 
lower platform. Convenience requires that the rise from the 
platform to the lowest step should not be greater than the rise 
of the car steps. This rise is variable, but with the figures usually 
employed the application of the rule will make the platform 
5'' to 15'' above the rail. 

Position with respect to tracks. — Low platforms are gen- 
erally built to the ends of the ties, or, if at the level of the top 
of the rail, are built to the rail head. Car steps usually 
extend 4' 6" from the track center and are 14" to 24'' above the 
rail. The platform must have plenty of clearance, and when 
the platform is high its edge is generally required to be 5' 6" 
from the track center. 

287. Minor stations. For a complete discussion of the desigh 
of stations of all kinds, including the details, the student is re- 
ferred to ''Buildings and Structures of American Railroads/' 
by Walter G. Berg, once Chief Engineer of the Lehigh Valley 
Railroad. The subject is too large for adequate discussion here, 
but a few fundamental principles will be referred to. 

Rooms required. An office and waiting-room is the mini- 
mum. A baggage-room, toilet-rooms, and express office are 
successively added as the business increases. In the Southern 
States a separate waiting-room for colored people is generally 
provided. It used to be common to have separate waiting-rooms 
for men and women. Experience proved that the men's wait- 
ing-room became a lounging place and smoking-room for loafers, 
and now large single waiting-rooms are more common even in 
the more pretentious designs, smoking being excluded. The 
office usually has a bay window, so that a more extended view 



334 RAILROAD CONSTRUCTION. § 287* 

of the track is obtainable. The women's toilet-room is entered 
from the waiting-room. The men's toilet-room, although built 
immediately adjoining the other in order to simplify the plumb- 
ing, is entered from outdoors. Old-fashioned designs built the 
station as a residence for the station-agent; later designs have 
very generally abandoned this idea. ^^Combination" stations 
(passenger and freight) are frequently built for small local 
stations, but their use seems to be decreasing and there is now a 
tendency to handle the freight business in a separate building. 

288. Section-houses. These are houses built along the right- 
of-way by the railroad company as residences for the trackmen. 
The liability of a wreck or washout at any time and at any part 
of the road, as well as the convenience of these houses for ordinary 
track labor, makes it all but essential that the trackmen should 
live on the right-of-way of the road, so that they may be easily 
called on for emergency service at any time of day or night. 
This is especially true when the road passes through a thinly 
settled section, where it would be difficult if not impossible to 
obtain suitable boarding-places. It is in no sense an extrava- 
gance for a railroad to build such houses. Even from the direct 
financial standpoint the expense is compensated by the corre- 
sponding reduction in wages, which are thus paid partly in free 
house rent. And the value of having men on liand for emergen- 
cies will often repay the cost in a single night. Where the coun- 
try is thickly settled the need for such houses is not so great, and 
railroads will utilize or perhaps build any sort of suitable house, 
but on Southern or Western roads, where the need for such 
houses is greater, standard plans have been studied with great 
care, §0 as to obtain a maximum of durability, usefulness, com- 
fort, and economy of construction. (See Berg's Buildings, etc., 
noted above.) On Northwestern roads, protection against cold 
and rain or snow is the chief characteristic; on Southern roads 
good ventilation and durability must be chiefly considered. 
Such houses may be divided into two general classes — (a) those 
which are intended for trackmen only and which may be built 
with great simplicity, the only essential requirements being a 
living-room and a dormitory, and (b) those which are intended 
for families, the houses being then distinguished as "dwelling- 
houses for employees. 

289. Engine-houses. Small engine-houses are usually built 
rectangular in plan. Their minimum length should be some- 



§ 289. MISCELLANEOUS STRUCTURES AND BUILDINGS. 335 

what greater than that of the longest engine on the road. They 
may be built to accommodate two engines on one track, but 
then they should be arranged to be entered at either end^ so that 
neither engine must wait for the other. In .width there may be 
as many tracks as desired, but if the demand for stalls is large, 
it will probably be preferable to build a " roundhouse. '^ Rect- 
angular engine-houses are usually entered by a series of parallel 
tracks switching off from one or more main tracks, no turn-table 
being necessary. If a turn-table is placed outside (because one 




Fig. 162. — Engine-house. 



is needed at that part of the road) enough track should be allowed 
between the house and the turn-table so that engines may be 
quickly removed from the engine-house in case of fire Tvdthout 
depending on the turn-table to get them out of danger. 

Roundhouses. The plan of these is generally polygonal 
rather than circular. The straight walls are easier to build; the 
construction is more simple, and the general purpose is equally 
well served. They may be built as a part of a circle or a com- 
plete circle, a passageway being allowed, so that there are two 
entrances instead of one. When space is very Hmited a round- 
house with turn-table will accommodate more engines in pro- 
portion to the space required (including the approaches) than a 
rectangular house. The enlarged space on the outer side of each 
segment of a roundhouse furnishes the extra space which is needed 
for the minor repairs which aie usually made in a roundhouse. 
One disadvantage is that supervision is not quite so easy or effec- 



336 RAILROAD CONSTRUCTION. § 289. 

tive as in rectangular houses. Of course such houses are used 
not only for storing and cleaning engines, but also for minor 
repairs which do not require the engine to be sent to the shops 
for a general overhauling. 

Construction. The outer walls are usually of brick. The 
inner walls consist almost entirely of doors and the piers between 
them, although there is usually a low wall from the top of the 
door frames to the roof line, which usuall}^ slopes outward so as 
to turn rain-water away from the central space. 

Roofs. Many roofs have been built of slate with iron truss 
framing, with the idea of maximum durability. The slate is good, 
but experience shows that the iron framing deteriorates very 
rapidly from the action of the gases of combustion of the engines 
which must be ''fired" in the houses before starting. Roof 
frames are therefore preferably made of wood. 

Floors. These are variously constructed of cinders, wood, 
brick, and concrete. Brick has been found to be the best ma- 
terial. Anything short of brick is a poor economy; concrete is 
very good if properly done but is somewhat needlessly expensive. 

Ventilation. This is a troublesome more expensive. 
The general plan is to have ''smoke-jacks" which drop down 
over the stack of each engine as it reaches its precise place in its 
stall and which will carry away all smoke and gas. Such a 
movable stack is mdst easily constructed of thin metal — say 
galvanized iron — but these will be corroded by the gases of 
combustion in two or three years. Vitrified pipe, cast iron, 
expanded metal and cement, and even plain wood painted with 
"fireproof" paint, have been variously tried, but all methods 
have their unsatisfactory features. (For an extended discus- 
sion of roundhouse floors and ventilation see the Proc. Assoc, 
of Railway Supts. of Bridges and Buildings for 1898, pp. 112-135.) 

SNOW STRUCTURES. 

290. Snow-fences. Snow structures are of two distinct 
kinds — fences and sheds. A snow-fence implies drifting snow — 
snow carried by wind — and aims to cause all drifting snow to be 
deposited away from the track. Some designs actually succeed 
in making the wind an agent for clearing snow from the track 
where it has naturally fallen. A snow-fence is placed at right 
angles to the prevailing direction of the wind and 50 to 100 feet 
awa^ from the tracks. When the road line is at right angles to 



§ 291. MISCELLANEOUS STRUCTURES AND BUILDINGS. 337 

the prevailing wind, the right-of-way fence may be built as a 
snow-fence — high and with tight boarding. Hedges have some- 
times been planted to serve this purpose. When the prevailing 
wind is oblique, the snow fences must be built in sections where 
they will serve the best purpose. The fences act as wind break- 
ers, suddenly lowering the velocity of the mnd and causing the 
snow carried by the wind to be deposited along the fence. 
Portable fences are frequently used, which are placed (by per- 
mission of the adjoining property owners) outside of the right- 
of-way. If a drift forms to the height of the portable fence the 
fence may be replaced on the top of the drift, where it may act 
as before, forming a still higher drift. When the prevailing 
wind runs along the track line, snow-fences built in short sec- 
tions on the sides T\dll cause snow to deposit around them 
while it scours its way along the track line, actually clearing 
it. Such a method is in successful operation at some places on 
the White Mountain and Concord divisions of the Boston & 
Maine Railroad. Snow-fences, in connection with a moderate 
amount of shoveling and plowing, suffice to keep the tracks 
clear on railroads not troubled with avalanches. In such cases 
snow-sheds are the only alternative. 

291. Snow-sheds. These are structures which will actually 
keep the tracks clear from ^now regardless of its depth outside. 
Fortunately they are only necessary in the comparatively rare 
situations where the snowfall is excessive and where the snow 
is liable to slide doT\Ti steep mountain slopes in avalanches. 
These avalanches frequently bring dovm. T\dth them rocks, trees, 
and earth, which would otherwise choke up the road-bed and 
render it in a moment utterly impassable for weeks to come. 
The sheds are usually built of 12" X 12" timber framed in about 
the same manner as trestle timbering; the "bents" are some- 
times placed as close as 5 feet, and even this has proved insuffi- 
cient to withstand the force of avalanches. The sheds are there- 
fore so designed that the avalanche will be deflected over them 
instead of spending its force against them. Although these 
sheds are only used in especially exposed places, yet their length 
is frequently very great and they are liable to destruction by fire. 
To confine such a fire to a limited section, "fire-breaks" are 
made — i.e., the shed is discontinued for a length of perhaps 100 
feet. Then, to protect that section of track, a V-shaped de- 
flector will be placed on the uphill side which will deflect all 



338 



RAILROAD CONSTRUCTION. 



§291. 



descending material so that it passes over the sheds. Solid crib 
work is largely used for these structures. Fortunately suitable 
timber for such construction is usually plentiful and cheap 
where these structures are necessary. Sufficient ventilation 
is obtained by longitudinal openings along one side immediately 
under the roof. "Summer^' tracks are usually built outside 
the sheds to avoid the discomfort of passing through these semi- 
tunnels in pleasant weather. The fundamental elements in 
the design of such structures is shown in Fig. 163, which illus- 
trates some of the sheds used on the Canadian Pacific Railroad. 




^^ 



Level-fall shed 



Fig. 163. — Snow-sheds — Canadian Pacific Railroad, 



292. Turn-tables. The essential feature of a turn-table is a 
carriage of sufficient size and strength to carry a locomotive, 
the carriage turning on a pivot of sufficient size to carry such a 
load. The carriage revolves in a circular pit whose top has 
the same general level as the surrounding tracks. The car- 
riages were formerly made largely of wood; very many of 
those still in use are of cast iron. Structural steel is now uni- 



§ 292. MISCELLANEOUS STRUCTURES AND BUILDINGS. 339 

versally employed for all modern work and since the construc- 
tion of the carriage and the piA^ot is a special problem in struc- 
tures, no further attention mil here be paid to the subject, 
except to that part which the railroad engineer must work out . 
— laying out the site and preparing the foundation. The 
minimum length of such a carriage (and therefore the diameter 
of the pit) is evidentl}^ the length over all of the longest engine 
and tender in use on the road. Usually 60-foot turn-tables 
will suffice for an ordinary road, and for hght-traffic roads 
employing small engines, 50 feet or even less may be sufficient. 
Many of the heavier freight engines of recent make have a total 
length of 65 to 70 feet; therefore 75-foot turn-tables are a 
better standard for heavy-traffic roads. A retaining-wall 
should be built around the pit. The stabihty of this wall imme- 
diately under the tracks should be especially considered. The 
most important feature is the stability of the foundation of the 
pivot, which must sustain a concentrated pressure, more or less 
eccentric, of perhaps 300 tons. When firm soil or rock may 
be easily reached, this need give no trouble, but in a soft, treach- 
erous soil a foundation of concrete or piling may be necessary. 
If the soil is very porous, it may be depended on to carry away 
all rain-water which may fall into the pit before the foundations 
are affected, but when the soil is tenacious it may be necessary 
to drain the subsoil thoroughly and carry off immediately all 
surface drainage by means of subsoil pipes which have a suit- 
able outfall. 

The location of the turn-table in the yard is a part of the 
general subject of " Yards, '^ and will be considered in the next 
chapter. 



CHAPTER XIII. 
YARDS AND TERMINALS, 

293. Value of proper design. A large part of the total cost of 
handling traffic, particularly freight, is that incurred at terminals 
and stations. In illustration of this, consider the relative total 
cost of handling a car-load of coal and a car-load (of equal 
weight) of mixed merchandise. The coal will be loaded in 
bulk on the cars at the mines, where land is comparatively- 
cheap, and the cars grouped into a train without regard to order, 
since they are (usually) uniform in structure, loading, and con- 
tents. When the terminal or local station is reached they are 
run on tracks occupying property which is usually much cheaper 
than the site of the terminal tracks and freight-houses; they are 
unloaded by gravity into pockets or machine conveyors and the 
empty cars are rapidly hauled by the train-load out of the way. 
On the other hand, the merchandise is loaded by hand on the car 
from a freight-house occupying a central and valuable location, 
the car is hauled out into a yard occupying valuable ground, is 
drilled over the yard tracks for a considerable aggregate miJeage 
before starting for its destination, where the same process is re- 
peated in inverse order. In either case the terminal expenses are 
evidently a large percentage of the total cost and, once loaded, 
it makes but little difference just how far the car is hauled to the 
other terminal. But the very evident increase in terminal charges 
for general merchandise over those for coal (large as they are) 
gives a better idea of the magnitude of terminal charges. 

Many yards are the result of growth, adding a few tracks at a 
time, without much evidence of any original plan. In such 
cases the yard is apt to be very inefficient, requiring a much 
larger aggregate of drilling to accomplish desired results, requir- 
ing much more time and hence blocking traffic and finally adding 
greatly to the cost of terminal service, although the fact of its 
being a needless addition to cost may be unsuspected or not fully 
appreciated. An unwillingness or inability to spend money for 

340 



§ 295. YARDS AND TERMINALS. 341 

the necessary changes, and the difficulty of making the changes 
while the yard is being used, only prolong the bad state of 
affairs and an inefficient makeshift is frequently adopted. As- 
sume that an improvement in the design of the yard will permit 
a saving of the use of one s^^dtching engine, or for example, that 
the work may be accomplished with three switching engines in- 
stead of four. Assuming a daily cost of $25, we have in 313 
working days an annual saving of $7825, which, capitalized at 
5%, gives $156,500, enough to reconstruct any ordinary yard.* 

294. Divisions of the subject. The subject naturally divides 
itself into three heads — (a) Yards for receiving, classifying, and 
distributing freight cars, called more briefly freight yards; (b) 
yards and conveniences for the care of engines, such as ash tracks, 
turn-tables, coal-chutes, sand-houses, water-tanks, or water 
stand-pipes, etc., and (c) passenger terminals. 

FREIGHT YARDS. 

295. General principles. It should be recognized at the start 
that at many places an ideally perfect yard is impossible, or at 
least impracticable, generally because ground of the required 
shape or area is practically unobtainable. But there are some 
general principles which may and should be followed in every yard 
and other ideals which should be approached as nearly as possi- 
ble. Nevertheless every yard is an independent problem. Be- 
fore taking up the design of freight yards, it is first necessary to 
consider the general object of such yards and the general princi- 
ples by which the object is accomplished. These may be briefly 
stated as follows : 

1. A yard is a device, a machine, by which incoming cars are 
sorted and classified — some sent to warehouses for unloading, 
some sent to connecting railroads, some made up for local dis- 
tribution along the road, some sent for repairs, and, in short a 
device by which all cars are sent through and out of the yard as 
quickly as possible. 

2. Except when a road's business is decreasing, or when its 
equipment is greater than its needs and its cars must be stored, 
efficiency of management is indicated by the rapidity with which 
the passage of cars through the yard is accomplished. 

3. When^a yard is the terminal of a ''division," the freight 

* Estimate of Mr. H, G. Hetzler, C, B. & Q. Ry. 



342 RAILROAD CONSTRUCTION. § 295. 

trains will be pulled into a "receiving track" and the engine and 
caboose detached. The caboose will be run on to a "caboose 
track," which should be conveniently near, and the engine is run 
off to the engine yard. If the train is a " through" train and no 
change is to be made in its make-up, it will only need to wait for 
another engine and perhaps another caboose. If the cars are to 
be distributed, they will be drawn off by a switching engine to 
the "classification yard." 

4. The design of a yard is best studied by first picking out the 
ladder tracks and the through tracks which lead from one divi- 
sion of the yard to another. These are tracks which must always 
be kept open for the passage of trains, in contradistinction to 
the tracks on which cars may be left standing, even though it is 
only for a few moments, while drilling is being done. Such a set 
of tracks, which may be called the skeleton of the yard, is shown 
by heavy lines in Fig. 164. Each line indicates a pair of rails. 
The tracks of the storage yards are shown by the lighter lines. 

5. There is a distinct advantage in having all storage tracks 
double-ended — except "team tracks." Team tracks are those 
which have spaces for the accommodation of teams, so that load- 
ing or unloading may be done directly between the cars and teams. 
To avoid the necessity of teams passing over the tracks, these are 
best placed on the outskirts of the yard and consist of short stub- 
sidings arranged in pairs. But storage tracks should have an 
outlet at each end so as to reduce the amount of drilling neces 
sary to reach a car which may be at the extreme end of a long 
string of cars. This is done usually by means of two "ladder" 
tracks, parallel to each other, which thus make the storage 
tracks between them of equal length. 

6. The equality of length of these storage tracks is a point in- 
sisted on by many, but on the other hand, trains are not always of 
uniform length even on any one division. Loaded trains and 
trains of empties will vary greatly in length, and the various 
styles and weights of freight engines employed necessitate other 
variations in the weights and lengths of trains hauled. With 
storage tracks of somewhat variable length a larger percentage 
of track length may be utilized, there will be less hauling over a 
useless length of track, and (assuming that the plot of ground 
available for yard purposes has equally favorable conditions for 
yard design) more business may be handled in a yard of given 
area. 



§295. 



YARDS AND TERMINALS. 



343 




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344 RAILROAD CONSTRUCTION. § 295. 

7 Yards are preferably built so that the tracks have a grade 
of 0.5% — sometimes a little more than this — in the direction of 
the traffic through the yard. This grade, which will overcome a 
tractive resistance of 10 pounds per ton, will permit cars to be 
started down the ladder tracks by a mere push from the switch- 
ing engine. They are then switched on to the desired storage 
track and run down that track by gravity until stopped at the 
desired place by a brakeman riding on the cars 

8. Although not absolutely necessary, there is an advantage 
in having all frog numbers and switch dimensions imiform. 
No. 7 frogs are most commonly used. Sharper- angled frogs 
make easier riding, less resistance and less chance of derailment, 
but on the other hand require longer leads and more space. No. 
6 and even No. 5 frogs are sometimes used on account of economy 
of space, but they have the disadvantages of greater tractive re- 
sistance, greater wear and tear on track and rolling stock, and 
greater danger of derailment. 

296. Relation of yard to main tracks. Safety requires that 
there should be no connection between the yard tracks and the 
main tracks except at each end of the yard, where the switches 
should be amply protected by signals. Sometimes the main 
tracks run through the yard, making practically two yards— -one 
for the traffic in either direction — but this either requires a double 
layout of tracks and houses (such as ash tracks, coal-chutes, sand- 
houses, etc.), or a very objectionable amount of crossing of the 
main- line tracks. The preferable method is to have the main hne 
tracks entirely on the outside of the yard. A method which is in 
one respect still better is to spread the main tracks so that they 
run on each side of the yard. In this case there is never any 
necessity to cross one main track to pass from the yard to the 
other main track; a train may pass from the yard to either 
main track and still leave the other main track free and open. 
The ideal arrangement is that by which some of the tracks cross 
over or under all opposing tracks. By this means all connections 
between the 3^ard and the main tracks maybe by ''trailing" 
switches; that is, trains will run on to the main track in the 
direction of motion on that main track. Of course all this 
applies only to double main track. 

An important element of yard design is to have a few tracks im- 
mediately adjoining the main tracks and separate from the yard 
proper on which outgoing trains may await their orders to take 



296. 



YARDS AND TERMINALS. 



345 




346 RAILRQAD CONSTRUCTION. § 297. 

the main track. When the orders come, they may start at once 
without any delay, without interfering with any yard operations, 
and they are not occupying tracks which may form part of the 
system needed for switching. 

297. Minor freight yards. The term here refers to the sub- 
stations, only found in the largest cities, to which cars will be sent 
to save in the amount of necessary team hauling and also to re- 
lieve a congestion of such loading and unloading at the main 
freight terminal. The cars are brought to these yards sometimes 
on floats (as is done so extensively at various points around New 
York Harbor), or they are run down on a long siding running 
perhaps through the city streets. But the essential feature of 
these yards is the maximum utilization of every square foot of 
yard space, w^hich is always very valuable and which is frequently 
of such an inconvenient shape that a great ingenuity is required 
to obtain good results. There is generally a temptation to use 
excessively sharp curves. When the radii are greater than 175 
feet no especial trouble is encountered. Curves with radius as 
short as 50 feet have been used in some yards. On such curves 
the long cars now generally used make a sharper angle with each 
other than that for which the couplers w^ere designed and spe- 
cial coupler-bars become necessary. The two general methods 
of construction are (a) a series of parallel team tracks (as pre- 
viously described and as illustrated further in Fig. 165), and (6) 
the '4oop system,'' as is illustrated in Fig. 166. 

298. Transfer cranes. These are almost an essential feature 
for yards doing a large business. The transportation of built- 
up girders, castings for excessively heavy machinery, etc., which 
weigh five to thirty tons and even more, creates a necessity for 
machinery which will easily transfer the loads from the car to 
the truck and mce versa. An ordinary ^^ gin-pole" will serve the 
purpose for loads which do not much exceed five tons. A fixed 
framework, covering a span long enough for a car track and a 
team space, with a trolley traveling along the upper chord, is the 
next design in the order of cost and convenience. Increasing 
the span 'so that it covers two car tracks and two team spaces 
will very materially increase the capacity. Making the frame 
movable so that it travels on tracks which are parallel to the 
car tracks, giving the frame a longitudinal motion equal to two 
or three car lengths, and finally operating the raising and travel- 
ing mechanism by power, the facility for rapidly disposing of 



YARDS AND TERMINALS. 



347 




East 135th St. 

Fig. 166.— Minor Freight Yard on a Harbor Front. 



348 RAILROAD CONSTRUCTION. § 299 

heavy articles of freight is greatly increased. Of course only a 
very small proportion of freight requires such handling, and the 
business of a yard must be large or perhaps of a special character 
to justify and pay for the installation of such a mechanism. 
Figs. 165 and 166 each indicate a transfer crane, evidently of the 
fixed type. 

299. Track scales. The location of these should be on one of 
the receiving tracks near the entrance to the yard, but not on the 
main track. It is always best to have a ^^dead track" over the 
scales — i.e., a track which has one rail on the solid side wall of 
the scale pit and the other supported at short intervals by posts 
which come up through the scale platform and yet do not touch 
it. These rails and the regular scale rails switch into one track 
by means of point rails a few feet beyond each end of the scales. 
The switches should be normally set so that all trains will use 
the dead track, unless the scales are to be operated. It has been 
found possible in a gravity yard to weigh a train with very little 
loss of time by running each car slowly by gravity over the 
scales and weighing them as they pass over. 

ENGINE YARDS. 

300. General principles. Engine yards must contain all the 
tracks, buildings, structures, and facilities which are necessary 
for the maintenance, care, and storage of locomotives and for pro- 
viding them with all needed supplies. The supplies are fuel, 
water, sand, oil, waste, tallow, etc. Ash-pits are generally neces- 
sary for the prompt and economical disposition of ashes; engine- 
houses are necessary for the storage of engines and as a place 
where minor repairs can be quickly made. A turn-table is an- 
•ther all but essential requirement. The arrangement of all 
^hese facilities in an engine yard should properly depend on the 
form of the yard. In general they should be grouped together 
and should be as near as possible to the place where through en- 
gines drop the trains just brought in and where they couple on 
to assembled outgoing trains, so that all unnecessary running light 
may be avoided. In Figs. 164 and 167 are shown two designs 
which should be studied with reference to the relative arrange- 
ment of the vard facilities. 



ii 



§300. 



YARDS AND TERMINALS. 



349 




Fia. 167. — Engine Yard and Shops, XJrbana, III. 



350 RAILROAD CONSTRUCTION. § 300. 



PASSENGER TERMINALS. 

(Passenger terminals are one of the logical subdivisions of 
this chapter, but their construction does not concern one engineer 
in a thousand. The local conditions attending their construction 
are so varied that each case is a special problem in itself — a prob- 
lem which demands in many respects the services of the archi- 
tect rather than the engineer. The student who wishes to pursue 
this subject is referred to an admirable chapter iir " Buildings and 
Structures of American Railroads," by Walter G. Berg, Chief 
Engineer of the Lehigh Valley Railroad.) 




To face p. 350.) (Published through courtesy of Union Switch and Signal Co.) 



CHAPTER XIV. 

BLOCK SIGNALING. 
GENERAL PRINCIPLES. 

301. Two fundamental systems. The growth of systems of 
block signaling has been enormous Tvdthin the last few years — 
both in the amount of it and in the development of greater per- 
fection of detail. The development has been along tw^o general 
lines: (a) the manual, in which every change of signal is the re- 
sult of some definite action on the part of some signalman, bat in 
which every action is so controlled or limited or subject to 
the inspection of others that a mistake is nearly, if not quite, 
impossible; (h) the automatic^ in which the signals are oper- 
ated by mechanism, which cannot set a wrong signal as long as the 
mechanism is maintained in proper order. The fundamental 
principles of the two systems will be briefly outlined, after 
which the chief details of the most common systems will be 
pointed out. 

302. Manual systems. Any railroad which has a telegraph 
line and an operator at all regular stations may (and frequently 
does) operate its trains according to the fundamental princi- 
ples of the manual block system even though it makes no claim 
to a block-signal system. The basic idea of such a system is 
that after a train has passed a given telegraph- or signal-station, 
no other train vn\l be permitted to follow it into that "block" 
until word is telegraphed from the next station ahead that the 
first train has passed out of that block. With a double-track 
road the operation is very simple; trains may be run at short 
intervals w^th long blocks; with an average speed of 30 miles 
per hour and blocks 5 miles long, trains could be run on a ten 
minute interval (nearly). A road with any such traffic would, 
of course, have much shorter blocks, and, practically, they 
would need to be considerably shorter. 

With a single-track road the operation is much more complex, 
since the operator must keep himself informed of the move- 

351 



352 RAILROAD CONSTRUCTION. § 302. 

ments of the trains in both directions. The ratio of length of 
block to train interval would be onl}^ one half (and practically 
much less than half) what it could be with a double-track road> 
When such a system is adhered to rigidly, it is called an absolute 
block system But when operating on this system, a delay of 
one train will necessarily delay every other train that follows 
closely after. A portion, if not all, of the delay to subsequent 
trains may be avoided, although at some loss of safety, by a 
system of permissive blocking. By this system an operator 
may give to a succeeding train a "clearance card'' which per- 
mits it to pass into the next block, but at a reduced speed and 
with the train under such control that it may be stopped on 
very short notice, especially near curves. One element of the 
danger of this system is the discretionary power with which it 
invests the signalmen, a discretion which may be wrongfully 
exercised. A modification (which is a fruitful source of colli- 
sions on single-track roads) is to order two trains to enter a 
block approaching each other, and with instructions to pass 
each other at a passing siding at which there is no telegraph- 
station. When the instructions are properly made out and 
literally obeyed, there is no trouble, but every thousandth or 
ten thousandth time there is a mistake in the orders, or a mis- 
understanding or disobedience, and a collision is the result. The 
telegraph line, a code of rules, a corps of operators, and sig- 
nals under the immediate control of the operators, are all that 
is absolutely needed for the simple manual system. 

303. Development of the manual system. One great diffi- 
culty with the simple system just described is that each operator 
is practically independent of others except as he may receive 
general or specific orders from a train-dispatcher at the division 
headquarters. Such difficulties are somewhat overcome by a 
very rigid system of rules requiring the signalmen at each station 
to keep the adjacent signalmen or the train-dispatcher in- 
formed of the movements of all trains past their own stations. 
When these rules (which are too extensive for quotation here) 
are strictly observed, there is but little danger of accident, and 
a neglect by any one to observe any rule will generally be appar- 
ent to at least one other man. Nevertheless the safety of trains 
depends on each signalman doing his duty, and a little careless- 
ness or forgetfulness on the part of any one man may cause an 
accident. The signaling between stations may be done by 



§ 303. BLOCK SIGNALING. 353 

ordinary telegraphic messages or by telephone, but is frequently 
done by electric bells, according to a code of signals, since these 
may be readily learned by men who would have more difficulty 
in learning the Morse code. 

In order to have the signalmen mutually control each other, 
the "controlled manual" system has been debased. The first 
successful system of this kind which was brought into exten- 
sive use is the "Sykes'^ system, of which a brief description 
is as follows: Each signal is worked by a lever; the lever is 
locked by a latch, operated by an electro-magnet, which, mth 
other necessar}^ apparatus, is inclosed in a box. When a signal 
is set at danger, the latch falls and locks the lever, which cannot 
be again set free until the electro-magnet raises the latch. The 
magnet is energized only by a current, the circuit of which is 
closed by a "plunger" at the next station ahead; just above 
the plunger is an "indicator," also operated by the current, 
which displays the words clear or blocked. (There are varia- 
tions on this detail.) ^Yhen a train arrives at a block station 
(A), the signalman should have pre^dously signaled to the station 
ahead (B) for permission to free the signal. The man ahead (B) 
pushes in the "plunger" on his instrument (assuming that the 
previous train has already passed him), which electrically opens 
the lock on the lever at the pre\aous station (A). The signal 
at A can then be set at "safety." As soon as the train has 
passed A the signal at A must be set at " danger." A further 
development is a device by which the mere passage of the train 
over the track for a few feet beyond the signal will automati- 
cally throw the signal to "danger." After the signal once goes 
to danger, it is automatically locked and cannot be released 
except by the man in advance (B), who will not do so until the 
train has passed him. The "indicator" on 5's instrimient 
shows "blocked" when A's signal goes to danger after the train 
has passed A, and B's plunger is then locked, so that he can- 
not release A's signal while a train is in the block. As soon as 
the train has passed A, B should prepare to get his signals ready 
by signahng ahead to C, so that if the block between B and C 
is not obstructed, B may have his signals at "safety" so that 
the train may pass B without pausing. The student should 
note the great advance in safety made by the Sykes system; 
a signal cannot be set free except by the oombined action of 
two men, one the man who actually operates the signal and 



354 RAILROAD CONSTRUCTION. § 304. 

the other the man at the station ahead, who frees the signal 
electrically and who by his action certifies that the block im- 
mediately ahead of the train is clear. 

A still further development makes the system still more " auto- 
matic '' (as described later), and causes the signal to fall to dan- 
ger or to be kept locked at danger, if even a single pair of wheels 
comes on the rails of a block, or if a switch leading from a main 
track is opened. 

304. Permissive blocking. "Absolute'' blocking renders ac- 
cidents due to collisions almost impossible unless an engineer 
runs by an adverse signal. The signal mechanism is usually 
so designed that, if it gets out of order, it will inevitably fall to 
'^danger,'' i.e., as described later, the signal-board is counter- 
balanced by a weight which is much heavier. If the wire breaks, 
the counterweight will fall and the board will assume the hori- 
zontal position, which always indicates "danger." But it some- 
times happens that when a train arrives at a signal-station, the 
signalman is unable to set the signal at safety. This may be 
because the previous train has broken down somewhere in the 
next block, or because a switch has been left open, or a rail has 
become broken, or there is a defect of some kind in the electrical 
connections. In such cases, in order to avoid an indefinite 
blocking of the whole traffic of the road, the signalman may 
give the engineer a "caution-card" or a "clearance card," 
which authorizes him to proceed slowly and with his train under 
complete control into the block and through it if possible. If 
he arrives at the next station without meeting any obstruction 
it merely indicates a defective condition of the mechanism, 
which will, of course, be promptly remedied. Usually the next 
section will' be found clear, and the train may pro(Sped as usual. 
On roads where the "controlled manual" system has received 
its highest development, the rules for permissive blocking are 
so rigid that there is but little danger in the practice, unless 
there is an absolute disobedience of orders. 

305. Automatic systems. By the very nature of the case, 
such systems can only be used to indicate to the engineers of 
trains something with reference to the passage of previous 
trains. The complicated shifting of switches and signals which 
is required in the operation of yards and terminals can only be 
accomplished by "manual" methods, and the only automatic 
features of these methods consist in the mechanical checks 



§ 306. BLOCK SIGNALING. 355 

(electric and otherwise), which will prevent wrong combina- 
tions of signals. But for long stretches of the road, where it 
is only required to separate trains by at least one block length, 
an automatic system is generally considered to be more relia- 
ble. As expressed forcibly by a railroad manager, "an auto- 
matic system does not go to sleep, get drunk, become insane, 
or tell lies when there is any trouble." The same cannot always 
be said of the employes of the manual system. 

The .basic idea of all such systems is that when a train passes 
a signal-station (A), the signal automatically assumes the "dan- 
ger" position. This may be accomplished electrically, pneu- 
matically, or even by a direct mechanism. When the train 
reaches the end of the block at B and passes into the next one, 
the signal at B will be set at danger and the signal at A will be 
set at safet}^ The lengths of the blocks are usually so great 
that the only practicable method of controlling from B a 
mechanism at A is by electricity, although the actual motive 
power at A may be pneumatic or mechanical. At one time 
the current from A to B was carried on ordinary wires. This 
method has the very positive advantage of reliability, definite 
resistance to the current, and small probability of short-circuit- 
ing or other derangement. But now all such systems use the 
rails for a track circuit and tliis makes it possible to detect the 
presence of a single pair of wheels on the track anywhere in the 
block, or an open switch, or a broken rail. Any such circum- 
stances, as w^ell as a defect in the mechanism, will break or 
short-circuit the current and will cause the signal to be set at 
danger. To prevent an indefinite blocking of traffic owing to 
a signal persistently indicating danger, most roads employing 
such a system have a rule substantially as follows : When a train 
finds a signal at danger, after waiting one minute (or more, 
depending on the rules), it may proceed slowly, expecting to 
find an obstruction of some sort; if it reaches the next block 
without finding any obstruction and finds the next signal clear, 
it may proceed as usual, but must promptly report the case to 
the superintendent. Further details regarding thpse methods 
will be given later. See § 310. 

306. "Distant" signals. The close running of trains that 
is required on heavy-traffic roads, especially where several 
branches combine to enter a common terminal, necessitates the 
use of ver}^ short blocks. A heavy train running at high speed 



356 RAILROAD CONSTRUCTION. § 306. 

can hardly make a "service'' stop in less than 2000 feet, while 
the curves of a road (or other obstructions) frequentl}^ make 
it difficult to locate a signal so that it can be seen more than a 
few hundred feet away. It would therefore be impracticable 
to maintain the speed now used with heavy trains if the engi- 
neer had no foreknowledge of the condition in which he wilMj 
find a signal until he arrives within a short distance of it. To* 
overcome this difficulty the "distant'' signal was devised. This 
is placed about 1800 or 2000 feet from the "home" signal, and 
is interlocked with it so that it gives the same signal. The dis- 
tant signal is frequently placed on the same pole as the home 
signal of the previous block. When the engineer finds the 
distant signal "clear," it indicates that the succeeding home 
signal is also clear, and that he may proceed at full speed and 
not expect to be stopped at the next signal; for the distant 
signal cannot be cleared until the succeeding home signal is 
cleared, which cannot be done until the block succeeding that 
is clear. A clear distant signal therefore indicates a clear track 
for two succeeding blocks. When the engineer finds the distant 
signal blocked, he need not stop (providing the home signal is 
clear). It simply indicates that he must be prepared to stop 
at the next home signal and must reduce speed if necessary. 
It may happen that by the time he reaches the succeeding home 
signal it has already been cleared, and he maj^ proceed without 
stopping. This device facilitates the rapid running of trains, 
with no loss of safety, and yet with but a moderate addition to 
the signaling plant. 

307. "Advance'* signals. It sometimes becomes necessary 
to locate a signal a few hundred feet short of a regular passen- 
ger-station. A train might be halted at such a signal because 
it was not cleared from the signal-station ahead — perhaps a 
mile or two ahead. For convenience, an "advance" signal 
may be erected immediately beyond the passenger-station. 
The train will then be permitted to enter the block as far as 
the advance signal and may deliver its passengers at the station. 
The advance signal is interlocked with the home signal back 
of it, and cannot be cleared until the home signal is cleared and 
the entire block ahead is clear. In one sense it adds another, 
block, but the signal is entirely controlled from the signal statioii 
back of it. 



§ 308. _ BLOCK SIGNALING. 357 



MECHANICAL DETAILS. 

308. Signals. The primitive signal is a mere cloth flag. A 
better signal is obtained when the flag is suspended in a suit- 
able place from a fixed horizontal support, the flag weighted 
at the bottom, and so arranged that it may be drawn up and 
out of sight by a cord which is run back to the operator's office. 
The next step is the substitution of painted wood or sheet metal 
for the cloth flag, and from this it is but a step to the standard 
semaphore on a pole, as is illustrated in Fig. 168. The simple 
flag, operated for convenience with a cord, is the signal em- 
ployed on thousands of miles of road, where they perhaps make 
no claim to a block-signal system, and yet where the trains 
are run according to the fimdamental rules of the simple manual 
block method. 

Semaphore boards. These are about 5 feet long, 8 inches 
wdde at one end, and tapered to about 6 inches wide at the hinge 
end. The boards are fastened to a casting which has a ring to 
hold a red glass which may be swung over the face of a lantern, 
so as to indicate a red signal. '^Distant'' signal-boards usually 
have their ends notched or pointed; the '^home^' signal-boards 
are square ended. The boards_are always to the right of the 
hinge when a train is approaching them. The "home'' signals 
are generally painted red and the "distant" signals green, 
although these colors are not invariable. The backs of the 
boards are painted white. Therefore any signal-board which 
appears on the left side of its hinge will also appear whitej and 
is a signal for traffic in the opposite direction, and is therefore 
of no concern to an engineman. 

Poles and bridges. When the signals are set on poles, they 
are generall}^ placed on the right-hand side of the track. When 
there are several tracks, four or more, a bridge is frequently 
built and then each signal is over its own track. When switches 
rim off from a main track, there may be several signal-boards 
over one track. The upper one is the signal for the main track 
and the lower ones for the several switches. In Fig. 169 is 
shown a "bridge" mth its various signal-boards controlling the 
several tracks and the smtches running off from them. 

"Banjo" signals. This name is given to a form of signal, 
illustrated in Fig. 170, in which the indication is taken from the 



358 RAILROAD CONSTRUCTION, § 308. 

color of a round disk inclosed with glass. This is the distinctive 
signal of the Hall Signal Company, and is also used by the 
Union Switch and Signal Company. The great argument in 
their favor is that they may be worked by an electric current 
of low voltage, which is therefore easily controlled ; that the 
mechanism is entirely inside of a case, is therefore very light, 
and is not exposed to the weather. The argument urged 
against them is that it is a signal of color rather than fosrm 
or 'position, and that in foggy weather the signal cannot be 
seen so easily; also that unsuspected color-blindness on the 
part of the engineman may lead to an accident. Notwith- 
standing these objections, this form of signal is used on thousands 
of miles of line in this country. 

309. Wires and pipes. Signals are usually operated by levers 
in a signal-cabin, the levers being very similar to the reversing- 
lever of a locomotive. The distance from the levers to the sig- 
nals is, of course, very variable, but it is sometimes 2000 feet. 
The connecting-link for the most distant signals is usually 
No. 9 wire; for nearer signals and for all switches operated 
from the cabin it may be 1-inch pipe. When not too long, one 
pipe will serve for both motions, forward and back. When 
wires are used, it is sometimes so designed (in the cheaper sys- 
tems) that one wire serves for one motion, gravity being de- 
pended on for the other, but now all good systems require two 
.wires for each signal. 

Compensators. Variations of temperature of a material with 
as high a coefficient as iron will cause very appreciable differ- 
ence of length in a distance of several hundred feet, and a 
dangerous lack of adjustment is the result. To illustrate: A 
fall of 60° F. will change the length of 1000 feet of wdre by 

1000 X 60 X. 0000065 =0.39 foot = 4.68 inches. 

A much less change than this will necessitate a readjustment 
of length, unless automatic compensators are used. A com- 
pensator for pipes is very readily made on the principle illus- 
trated in Fig. 171. The problem is to preserve the distance 
between a and d constant regardless of the temperature. Place 
the compensator half-way between a and d, or so that ah=cd. 
A fall of temperature contracts ah to ah\ Moving h to h' will 
cause c to move to c', in which hb^ = cc\ But cd has also short- 
ened to c'd; therefore d remains fixed in position. 



(To {ace page 358.) 




Courtesy of the Z'n/on ^Rv?r/i and Signal Co. 

Fig. 168. — Semaphores. 



I 




{To face 'page 3S8.) 




Courtesy of the Union Switch and Signal OOk 

Fig. 170. — " Banjo " Signals. 



§309. 



BLOCK SIGNALING. 



359 



The regulations of the A. R. E. & M. W. Assoc, require that 
"A compensator shall be provided for each pipe line over fifty 
(50) feet in length and under eight hundred (800) feet, with 
crank-arms eleven by thirteen (11X13) inch centers. From 
eight hundred (800) to twelve hundred (1200) feet in length, 
crank-arms shall be eleven by sixteen (11X16) inch centers. 
Pipe lines over twelve hundred (1200) feet in length shall be 
provided with an additional compensator. 

'^Compensators shall have one sixty (60) degree and one one 
hundred and twenty (120) degree angle-cranks and connecting 




Fig. 171. — Standard Pipe Compensator. 



link, mounted in cast iron base, having top of center pins sup- 
ported. The distance between center of pin-holes shall be 
twenty-two (22) inches." 

The compensator should be placed in the middle of the length 
when only one is used. "Wlien two are used they should be 
placed at the quarter points. Note that in operating through 
a compensator the direction of motion changes; i.e., if a moves 
to the right, d moves to the left, or if there is compression in ah 
there is tension in cd, and vice versa. Therefore this form of 
compensator can only be used with pipes which will withstand 
compression. It has seemed impracticable to design an equally 



360 



RAILBOAD CONSTRUCTION, 



§309, 



satisfactory compensator for wires, although there are several 
designs on the market. 

The change of length of these bars is so great that allowance 
must be made for the temperature at the time of installation. 
On the basis of 50° as the mean temperature, the pipes are so 
adjusted that the distance between the points h and c of Fig. 171 
is made greater or less than 22 inches, according to the tem- 
perature of installation. For example, if the temperature were 
80° and the length of the piping were 900 feet, the length of the 
pipes should be adjusted so that be is less than 22 inches by an 
amount equal to 900 X (80°- 50°) X .0000065 = 0.1755 feet = 
2.106 inches. The length should therefore be 19.9 inches in- 
stead of 22 inches. If the mean temperature was very different 
(say in Florida) some higher temperature should be taken as 
normal, so that the extreme range above and below the normal 
shall be approximately the same. 

Guides around curves and angles. When wires are required 
to pass around curves of large angle, pulleys are used, and a 
length of chain is substituted for the wire. For pipes, when 
the curve is easy the pipes are slightly bent and are guided 
through pulleys. When the angle is sharper, '' angles" are 
used. The operation of these details is self-evident from an 
inspection of Fig. 172. 

310. Track circuit for automatic signaling. The several 
systems of automatic signaling differ in the minor details, but 
nearly all of them agree in the following particulars. A current 
of low potential is run from a battery at one end of a section 
through one line of rails to the other end of the section, then 
through a relay, and then back to the battery through the other 




Fig. 172. — Deflecting-rods. 



line of rails. To avoid the excessive resistance which would 
occur at rail joints which may become badly rusted, a wire 



§310. 



BLOCK SIGNALING. 



361 



suitably attached to the rails is run around each joint. In 
order to insulate the rails of one sec- _^ 

tion from the rails at either end and 
yet maintain the rails structurally con- 
tinuous, the ends of the rails at these 
dividing points are separated by an 
insulator and the joint pieces are either 
made of wood or have som3 insulating 
material placed between the rails and 
the ordinary metal joint. The bolts 
must also be insulated. When the 
relay is energized by a current, it 
closes a local circuit at the signal- 
station, which will set the signal there 
at "safety." Tha resistance of the 
relay is such that it requires nearly the 
whole current to work it and to keep 
the local circuit closed. Therefore, 
when there is any considerable loss of 
current from one rail to the other, the 
relay will not be sufSciently energized, 
the local circuit will be broken, and the 
signal will automatically fall to-danger. 
This diversion of current from one rail 
to the other before the current reaches 
the relay may be caused in several 
ways: the presence of a pair of wheels 
on the rails anywhere in the section will 
do it ; also the breakage of a rail ; also 
the opening of a switch anywhere in 
the section ; also the presence of a pair 
of wheels on a siding between the 
"fouling point'' and the switch. (The 
"fouling point'' of a siding is that 
point where the rails first commence to 
approach the main track.) In Fig. 173 
is shown all of the above details as well 
as some others. At A, B, and the 
"fouling point" are shown the in- 
sulated joints. The batteries and 
signals are arranged for train motion 




Fio. 173. 



362 RAILROAD CONSTRUCTION. § 310, 

to the right. When a train has passed the points near A, where 
the wires leave the rails for the relay, the current from the 
"track battery'' at B will pass through the wheels and axles, and 
although no electrical connection is broken, so much current 
will be shunted through tha wheels and axles that the weak 
current still passing through the relay is not strong enough to 
energize it against its spring and the " signal-magnet'' circuit 
is broken, and the signal A goes to "danger." At the turnout 
the rails between the fouling point and the switch are so con- 
nected (and insulated) that a pair of wheels on these rails will 
produce the same effect as a pair on the main track. This is 
to guard against the effect of a car standing too near the switch, 
even though it is not on the main track. When the train passes 
Bj if there is no other interruption of the current, the track 
battery at B again energizes the relay at A, the signal-magnet 
circuit at A is closed, and the signal is drawn to "safety." 

(The present edition has omitted several subdivisions of this 
general subject, notably the "staff system," used chiefly in 
England, and all discussions of "interlocking'' which is an 
essential feature of the operation of large terminal yards. A 
future edition may supply these deficiencies, although an ex- 
haustive treatment of the subject of Signaling would require a 
separate volume.) 



CHAPTER XV. 



ROLLING-STOCK. 



(It is perhaps needless to say that the following chapter is 
in no sense a course in the design of locomotives and cars. Its 
chief idea is to give the student the elements of the construc- 
tion of those vehicles which are to use the track which he may- 
design — to point out the mutual actions and reactions of vehicle 
against track and to show the effect on track wear of varia- 
tions in the design of rolling-stock. The most of the matter 
given has a direct practical bearing on track-work, and it is con- 
sidered that all of it is so closely related to his work that the 
civil engineer may study it with profit.) 



WHEELS AND RAILS. 

311. Effect of rigidly attaching wheels to their axles. The 

wheels of railroad rolling-stock are invariably secured rigidly 
to the axles, which therefore revolve wath the wheels. The 
chief reason for this is to avoid excessive wear 
between the axles and the w^heels. 

Any axle must always be somewhat loose in 
its journals. A sidewise force P (see Fig. 174) 
acting against the circumference of the w^heel 
will produce a much greater pressure on the 
axle at S and S% and if the wheel moves on 
the axle, the wear at S and S^ will be exces- 
sive. But when the axle is fitted to the wheel 
with a "forced fif and does not revolve, 
the mere pressure produced at S is harmless. 
When two wheels are fitted tight to an axle, 
as in Fig. 175, and the axle revolves in the jour- 
nals aa, a sidewise pressure of the rail against the wheel flange 
will only produce a slight and harmless increase of the journal 
pressure Q, although at Q there is sliding contact. Twist- 

363 ' 




364 



BAILROAD CONSTRUCTION. 



§ 312. 



ing action in the journals is thus practically avoided, since a 
small pressure at the journal-boxes at each end of the axle 
suffices to keep the axle truly in line. 




Fig. 175. 



Fig. 176. 



On the other hand, when the wheels are rigidly attached to 
their axles, both wheels must turn together, and when rounding 
curves, the inner rail being shorter than the outer rail, one 
wheel must slip by an amount equal to that difference of length. 
The amount of this slip is readily computable : 



Longitudinal slip = 



'360^ 



(r2 






--Ca'' 



(136) 



in which C is a constant for any one gauge, and g= the track 
gauge = (r2 — ri). For standard gauge (4.708) the shp is .08218 
foot per degree of central angle. This shows that the longitu- 
dinal slipping around any curve of any given central angle will 
be independent of the degree of the curve. The constant (.08218) 
here given is really somewhat too small, since the true gauge 
that should be considered is the distance between the lines of 
tread on the rails. This distance is a somewhat indeterminate 
and variable quantity, and probably averages 4.90 feet, which 
would increase the constant to .086. The slipping may occur 
by the inner wheel slipping ahead or the outer wheel slipping 
back, or by both wheels slipping. The total slipping will be 
constant in any case. The slipping not only consumes power, 
but wears both the wheels and the rail. But even these dis- 
advantages are not sufficient to offset the advantages resulting 
from rigid wheels and axles. 

312. Effect of parallel axles. Trucks are made with two or 
three parallel axles (except as noted later), in order that the 
axles shall mutually guide each other and be kept approximately 



§312. 



ROLLING-STOCK. 



365 



perpendicular to the rails. If the curvature is very sharp and 
the wheel-base comparatively long (as is notably the case on 
street railways at street corners), the front and rear wheels 





Fig. 178. 



- \ \ \ 



Fig. 179. 



will stand at the same angle (a) with the track, as shown in 
Fig. 177. But it has been noticed that for ordinary degrees of 
curvature, the rear wheels stand radial to the curve (see Fig. 
178), and for steam railroad work this is the normal case. When 
the two parallel axles are on a curve (as shown), the wheels tend 
to run in a straight line. In order that they shall run on a curve 
they must slip laterally. The principle 
is illustrated in an exaggerated form in 
Fig. 179. The wheel tends to roll from a j 
toward h. Therefore in passing along the 
track from a to c it must actually slip late- ''' 
rally an amount he which equals ae sin a. 
Let ^= length of the wheel-base (Figs. 177 and 178); r= radius 
of curve; then for the first case (Fig. 177), sin a = t-r-2r; for 
the second and usual case (Fig. 178), sin a = ^^r; ior t = 5 feet 
and r = radius of a 1° curve, a = 0°03' for the second case, a 
varies (practically) as the degree of curve. The lateral slipping 
per unit of distance traveled therefore equals sin a. As an 
illustration, given a 5-foot wheel-base on a 5° curve, a = 0° 15', 
sin a = .00436, and for each 100 feet traveled along the curve 
the lateral slip of the front wheels would be 0.436 foot. There 
would be no lateral slipping of the rear wheels, assuming that 
the rear axle maintained itself radial. 

From the above it might be inferred that the flanges of the 
forward wheels will have much greater w^ear than those of the 
rear wheels. Since cars are drawn in both directions about 
equally, no difference in flange wear due to this cause will occur, 
but locomotives (except switching-engines) run forward almost 



366 



BAILROAD CONSTRUCTION. 



§313. 



exclusively, and the excess wear of the front wheels of the pilot - 
and tender-trucks is plainly observable. 

For a given curve the angle a (and the accompanying resist- 
ance) is evidently greater the greater the distance between 
the axles. On the other hand, if the two axles are very close 
together, there will be a tendency for the truck to twist and 
the wheels to become jammed, especially if there is consider- 
able play in the gauge. The flange friction would be greater 
and would perhaps exceed the saving in lateral slipping. A 
general rule is that the axles should never be closer together 
than the gauge. 

Although the slipping per unit of length along the curve varies 
directly as the degree of curvature, the length of curve necessary 
to pass between two tangents is inversely as the degree of curve, 
and the total slipping between the two tangents is independent 
of the degree of curve. Therefore when a train passes between 

two tangents, the total slipping 
of the wheels on the rails, lon- 
gitudinal and lateral, is a quantity 
which depends only on the central 
angle and is independent of the 
radius or degree of curve. 

313. Effect of coning wheels. 
The wheels are always set on the 
axle so that there is some "play'' 
or chance for lateral motion be- 
tween the wheel-flanges and the 
rail. The treads of the wheel are 
also " coned." This coning and play 
of gauge are shown in an exagger- 
ated form in Fig. 180. When the 
wheels are on a tangent, although there will be occasional oscil- 
lations from side to side, the normal position will be the sym- 
metrical position in which the circles of tread bh are equal. 
When centrifugal force throws the wheel-flange against the rail, 
the circle of tread a is larger than b, and much larger than c; 
therefore the wheels Avill tend to roll in a circle whose radius 
equals the slant height of a cone whose elements would pass 
through the unequal circles a and c. If this radius equaled the 
radius of the track, and if the axle were free to assume a radial 
position, the wheels would roll freely on the rails without any 




Fig. 180. 



§ 314. ROLLING-STOCK. 367 

slipping or flange pressure. Under such ideal conditions, 
coning would be a valuable device, but it is impracticable to 
have all axles radial, and the radius of curvature of the track 
is an extremely variable quantity. It has been demonstrated 
that with parallel axles the influence of coning diminishes as 
the distance between the axle increases, and that the effect is 
practically inappreciable when the axles are spaced as they are 
on locomotives and car-trucks. The coning actually used is 
very slight (see Chapter XV, § 332) and has a different object. 
It is so slight that even if the axles were radial it would only 
prevent the slipping on a very light curve — say a 1° curve. 

314. Effect of flanging locomotive driving-wheels. If all the 
wheels of all locomotives were flanged it would be practically 
impossible to run some of the longer types around sharp curves. 
The track-gauge is always widened on curves, and especially 
on sharp curves, but the widening would need to be excessive 
to permit a consolidation locomotive to pass around an 8° or 
10° curve if all the drivers were flanged. The action of the 
wheels on a curve is illustrated in Figs. 181, 182, and 184. All 
small truck-wheels are flanged. The rear drivers are always 
flanged and four-driver engines usually have all the drivers 
flanged. Consolidation engines have only the front and rear 
drivers flanged. Mogul an^ ten-wheel engines have one pair 
of drivers blank. On Mogul engines it is always the middle 
pair. On ten-wheel engines, when used on a road having sharp 
curves, it is preferable to flange the front and rear driving- 
wheels and use a ''swing bolster '' (see § 315); when the curva- 
ture is easy, the middle and rear drivers may be flanged and 
the truck made with a rigid center. The blank drivers have 
the same total width as the other drivers and of course a much ^ 
wider tread, which enables these drivers to remain on the rail; 
even though the curvature is so sharp that the tread overhangs 
the rail considerably. 

315. Action of a locomotive pilot-truck. The purpose of 
the pilot-truck is to guide the front end of a locomotive around 
a curve and to relieve the otherwise excessive flange pressure 
that would be exerted against the driver-flanges. There are 
two classes of pilot -trucks — (a) those having fixed centers and 
(h) those having shifting centers. This second class is again 
subdivided into two classes, which are radically different in 
their action — (h{) four-wheeled trucks having two parallel axles 



368 



RAILROAD CONSTRUCTION. 



§315. 



and (62) two-wheeled trucks which are guided by a "radius- 
bar." The action of the four-wheeled fixed-centered truck (a) 
is shown in Fig. 181. Since the center of the truck is forced 




Fig. 181. — Fixed Center Pilot-truck. 
to be in the center of the track, the front drivers are drawn 
away from the outer rail. The rear outer driver tends to roll 
away from the outer rail rather than toward it, and so the effect 




Fig. 182. — Four-wheeled Truck — Shifting Center. 

of the truck is to relieve the driver-flanges of any excessive 
pressure due to curvature. The only exception to this is the 
case where the curvature is sharp. Then the front inner driver 
may be pressed against the inner rail, as indicated in Fig. 181. 

This limits the use of this type of 
wheel-base on the sharper curves. 
The next type — Q)^ four-wheeled 
trucks with shifting centers — is 
much more flexible on sharp 
curvature; it likewise draws the 
front drivers away from the outer 
rail. The relative position of the 
wheels is shown in Fig. 182, in 
Avhich c' represents the position 
of center-pin and c the displaced 
truck center. The structure and 
action of the truck is shown in 
Fig. 183. The "center-pin" (1) is 
supported on the "truck-bolster" (2), which is hung by the 
''links" (4) from the "cross-ties" (3). The links are therefore 




Fig. 183. — Action of Shifting 
Center. 



§315. 



ROLLING-STOCK. 



369 



in tension and when the wheels are forced to one side by the 
rails the links are incHned and the front of the engine is 
drawn inward by a force equal to the weight on the bolster 
times the tangent of the angle of inclination of the links. This 
assumes that all links are vertical when the truck is in the 
center. Frequently the opposite links are normally inclined to 
each other, w^hich somewhat complicates the above simple relation 
of the forces, although the general principle remains identical. 

The two-wheeled pilot-truck with shifting center is illus- 
trated in Fig. 184. The figure shows the facility with which 





Fig. 185. — Action of Two- 
WHEf:i.ED Truck. 



Fig. 184. — Two-wheeled Truck — Shifting Center. 
an engine vaih. long wheel-base may be made to pass around 
a comparatively sharp curve by omitting the flanges from the 
middle drivers and using this form of pilot-truck. As in the 
previous case, the eccentricity of 
the center of the truck relative 
to the center-pin induces a cen- 
tripetal force which draws the 
front of the engine inw^ard. But 
the swing-truck is not the only 
source of such a force. If the 
''radius-bar pin'' were placed at 0' (see Fig. 185), the truck- 
axle would be radial. But the radius-bar is always made some- 
what shorter than this, and the pin is placed at 0, a considerable 
distance ahead of 0', thus creating a tendency for the truck 
to run toward the inner rail and draw the front of the loco- 
motive in that direction. This tendency will be objectionably 
great if the radius-bar is made too short, as has been practically 
demonstrated in cases when the radius-bar has been subse- 
quently lengthened with a resulting improvement in the running 
of the engine. This type of pilot truck is used on both Mogul 
and Consohdation locomotives and explains why these long 
engines can so easily operate on sharp curves. 



370 



RAILROAD CONSTRUCTION. 



§316. 



LOCOMOTIVES. 
GENERAL STRUCTURE. 

316. Frame. The frame or skeleton of a locomotive con- 
sists chiefly of a collection of forged wrought-iron bars, aa 
shown in Figs. 186 and 187. These bars are connected at the 



■=^ 



B^3iF 



Fig. 186. — Engine -frame. 

front end by the "bumper'^ (c), which is usually made of wood. 
A little further back they are rigidly connected at hb by the 
cylinders and boiler-saddle. The boilers rest on the frames 
at aaaa by means of ^'pads/^ which are bolted to the fire-box, 
but which permit a free expansion of the boiler along the frame. 
This expansion is sometimes as much as jV'- ^^ ^ " con- 
solidation '' engine (frame shown in Fig. 187) it is frequently 




Fig. 187. — Engine-frame — Consolidation Type. 

necessary to use vertical swing-levers about 12" long instead 
of ''pads." The swinging of the levers permit all necessary 
expansion. At the back the frames are rigidly connected by 
the iron "foot-plate." The driving-axles pass through the 
"jaws" dddd, which hold the axle-boxes. The frame-bars 
have a width (in plan) of 3'' to 4''. The depth (at a) is about 
the same. Fig. 186 shows a frame for an "American" type 
of locomotive; Fig. 187 shows a frame for a '' Consolidation" 
type (see §323). 

317. Boiler. A boiler is a mechanism for transferring the 
latent heat of fuel to water, so that the water is transformed 
from cold water into high-pressure steam, which b}^ its expan- 
sion will perform work. The efficiency of the boiler depends 
largely on its ability to do its work rapidly and to reduce to 
a minimum -the waste of heat through radiation. The boiler 
contains a fire-box (see Fig. 188), in which the fuel is burned. 
The gases of consumption pass from the fire-box through the 
numerous boiler-tubes into the "smoke-box" S and out through 
the smoke-stack. The fire-box consists of an inner and outer 



§ 317. 



ROLLING-STOCK. 



371 



shell separated by a layer of water about 3" thick. The ex- 
posure of water-surface to the influence of the fire is thus very 
complete. The efficiency of this transferal of heat is somewhat 
indicated by the fact that, although the temperature of the 
gases in the fire-box is probably from 3000° to 4000° F., the 
temperature in the smoke-box is generally reduced to 500° to 




Fig. 188. — LocoMOTnrE-BoiLER. 



600° F. If the steam pressure is 180 lbs., the temperature of 
the water is about 380° F., and, considering that heat will not 
pass from the gas to the water unless the gas is hotter than the 
water, the water evidentl}^ absorbs a large part of the theo- 
retical maximum. Nevertheless gases at a temperature of 
600° F. pass out of the smoke-stack and such heat is utterly 
wasted. 

The tubes vary from If" to 2" , inside diameter, with a thick- 
ness of about O'MO to 0".12. The aggregate cross-sectional 
area of the tubes should be about one eighth of the grate area. 
The number will vary from 140 to 250, They are made as long 
as possible, but the length is virtually determined by the type 
and length of engine. 

318. Fire-box. The fire-box is surrounded by water on the 
four sides and the top, but since the water is subjected to the 
boiler pressure, the plates, which are about -/g" thick, must be 
stayed to prevent the fire-box from collapsing. This is easily 
accomplished over the larger part of the fire-box surface by 
having the outside boiler-plates parallel to the fire-box plates 
and separated from them by a space of about 3". The plates 
are then mutually held by ''stay-bolts.'' See Fig. 189. These 
are about y in diameter and spaced 4'' to 4J''. The jV' ^^le, 



372 



KAILBOAD CONSTRUCTION. 



§318. 



drilled 1^" deep, indicated in the figure, will allow the escape 
of steam if the bolt breaks just behind the plate, and thus calls 
attention to the break. The stay-bolts are turned down to a 
diameter equal to that at the root of the screw-threads. This 
method of supporting the fire-box sheets is used for the two 
sides, the entire rear, and for the front of the fire-box up to the 
boiler-barrel. The ^'furnace tube-sheet'' — the upper part of 
the front of the fire-box — is stayed by the tubes. But the top 
of the fire-box is troublesome. It must always be covered 
with water so that it will not be "burned'' by the intense heat. 
It must therefore be nearly, if not quite, flat. There are three 
general methods of accomplishing this. 




Fig. 189. 



Fig. 190. 



(a) Radial stays. This construction is indicated in Fig. 190. 
Incidentally there is also shown the diagonal braces for resist- 
ing the pressure on the back end of the boiler above the fire- 
box. It may be seen that the stays are not perpendicular to 
either the crown-sheet or the boiler-plate. This is objection- 
able and is obviated by the other methods. 

(b) Crown-bars. These bars are in pairs, rest on the side 
furnace-plates, and are further supported by stays. See Fig. 
191. 

(c) Belpaire fire-box. The boiler above the fire-box is rect- 
angular, with rounded corners. The stays therefore are per- 
pendicular to the plates. See Fig. 192. 

Fire-brick arches. ThcKse are used, as shown in Fig. 193, to 
force all the gases to circulate through the upper part of the fire- 



§318. 



ROLLING-STOCK. 



373 



box. Perfect combustion requires that all the carbon shall be 
turned into carbon dioxide, and this is facilitated by the 
forced circulation. 




m 




OOOCDO 

.oooo!ooo_ _ 
ooooooooooo' 

>0 00|0 00 00 00 

.oooooooooooc 
00000:0000000 
,00000000000 
oooooiooooooc 

>OOOOCT)000000 
pOOOOOOOOOO' 

'0 0000 00 




Water-tables. The same object is attained by using a water- 
table instead of a brick arch— as shown in Fig. 191. But it has 



374 



RAILROAD CONSTRUCTION. 



§319. 



the lurther advantages of giving additional heating-surface and 
avoiding the continual expense of maintaining the bricks. One 




Fig. 192. — "Belpaire" Fire-box. 
Half-section through AB. Half -section through CD. 

feature of the design is the use of a number of steam- jets 
which force air into the fire-box and assist the combustion. 




Fig. 1 93. — Fire-brick Arch. 




Fig. 194. — ^Wootten Fire-box. 



Area. Fire-boxes are usually limited in width to the prac- 
ticable width between the wheels — thus giving a net inside 
width of about 3 feet and a maxinmm length of 10 to 11 feet — 
this being about the maximum distance over which the firemen 
can properly control the fire. About 37 square feet is the 
maximum area obtainable except when the "Wootten'^ fire- 
box is used — illustrated in Fig. 194. Here the grate is raised 
above the driving-wheels and has (in the case shown) a width 
of 8' 0Y\ The fire-box area is over 76 square feet. Note that 
two furnace-doors are used. 

319. Coal consumption. No form of steam-boiler (except 
a boiler for a steam fire-engine) requires as rapid production 
of steam, considering the size of the boiler and fire-box, as a 



§ 319. rolling-stock:. 375 

locomotive. The combustion of coal per square foot of grate 
per hour for stationary boilers averages about 15 to 25 lbs. and 
seldom exceeds that amount. An ordinary maximum for a 
lofcomotive is 125 lbs. of coal per square foot of grate-area per 
hour, and in some recent practice 220 lbs, have been used. Of 
course such excessive amounts are wasteful of coal, because 
a considerable percentage of the coal will be blown out of the 
'smoke-stack unconsumed, the draft necessary for such rapid 
consumption being very great. The only justification of such 
rapid and wasteful coal consumption is the necessity for rapid 
production of steam. The best quality of coal is capable of 
evaporating about 14 lbs. of water per pound of coal, i.e., change 
it from water at 212° to steam at 212°; the heat required to 
change water at ordinary temperatures to steam at ordinary 
working pressure is (roughly) about 20% more. From 6 to 9 lbs. 
of water per pound of coal is the average performance of ordinary 
locomotives, the efficiency being less with the higher rates of 
combustion. Some careful tests of locomotive coal consump- 
tion gave the following figures: when the consumption of coal 
was 50 lbs. per square foot of grate-area per hour, the rate of 
evaporation was 8 lbs. of water per pound of coal. When the 
rate of coal consumption was raised to ISO, the evaporation 
dropped to 5 lbs. of water" per pound of coal. It has been 
demonstrated that the efficiency of the boiler is largely increased 
by an increased length of boiler-tubes. The actual consump- 
tion of coal per mile is of course an exceedingly variable quan- 
tity, depending on the size and type of the engine and also on 
the work it is doing — whether cHmbing a heavy grade with its 
maximum train-load or running easily over a level or dowTi 
grade. A test of a 50-ton engine, running without any train at 
about 20 to 25 miles per hour, showed an average consumption 
of 21 lbs. of coal per mile. Statistics of the Pennsylvania Rail 
road show a large increase (as might be expected, considering 
the growth in size of engines and weight of trains) in the aver- 
age number of poxuids of coal burned per ^ram-mile— some of 
the figures being 55 lbs. in 1863, 72 lbs. in 1872, and nearly 
84 lbs. in 1883. Figures are published showing an average 
consumption of about 10 lbs. of coal per passenger-car mile, 
and 4 to 5 lbs. per freight-car mile. But these figures are always 
obtained by dividing the total consumption per train-mile by 
the number of cars, the coal due to the weight of the engine 



S76 RAILROAD CONSTRUCTION. § 320. 

being thrown in Wellington developed a rule, based on the 
actual performance of a very large number of passenger-trains, 
that the number of pounds of coal per mile = 21.1 + 6.74 times 
the number of passenger-cars. The amount of coal assigned 
to the engine agrees remarkably with the test noted above 
For freight-trains the amount assigned to the engine should 
be much greater (since the engine is much heavier), and that 
assigned to the individual cars much less, although the great 
increase in freight-car weights in recent years has caused an 
increase in the coal required per car. 

320. Heating- surface. The rapid production of steam re- 
quires that the hot gases shall have a large heating-surface to 
which they can impart their heat. From 50 to 75 square feet 
of heating-surface is usually designed for each square foot of 
grate-area. A more recently used rule is that there should be 
from 60 to 70 square feet of tube heating-surface per square 
foot of grate-area for bituminous coal. 40 or 50 to 1 is more 
desirable for anthracite coal Almost the whole surface of 
the fire-box has water behind it, and hence constitutes heating- 
surface. Although this surface forms but a small part of the 
total (nominally), it is really the most effective portion, since 
the difference of temperature of the gases of combustion and 
the water is here a maximum, and the flow of heat is therefore 
the most rapid. The heating-surface of the tubes varies from 
85 to 93% of the total, or about 7 to 15 times the heating-sur- 
face in the fire-box. Sometimes the heating-surface is as much 
as 2300 square feet, but usually it is less than 2000, even for 
engines which must produce steam rapidly. 

A ''Mikado" type locomotive, wheel plan, 2-8-2 (see § 323), 
constructed in 1912, has cylinders 28" diameter, 32" stroke; 
drivers, 64" diameter; steam pressure, 180 lbs.; grate area, 
78 sq.ft.; heating surface, 4627 sq.ft.; besides a superheating 
surface of 961 sq. ft.; total weight of engine and tender, 
480,000 lbs.; weight of engine alone, 303,400 lbs.; weight on 
drivers, 231,000 lbs.; tractive power 60,000. There is about 
71 sq.ft. of heating surface per square foot of grate area. The 
adhesion ratio is 1: 3.85 or about 26%. 

Another rule used by designers is that the engine should 
have 1 sq. ft. of heating-surface for each 50 or 60 lbs. of weight, 
efficiency being indicated by a low weight. For the above 
engine the ratio is 54. 



§ 321. ROLLING-STOCK. 377 

321. Loss of efficiency in steam pressure. The effective 
work done by the piston is never equal to the theoretical energy 
contained in the steam withdrawn from the boiler. This is due 
chiefly to the following causes: 

(a) The steam is "wire- drawn," i.e., the pressure in the 
cylinder is seldom more than 85 to 90% of the boiler pressure. 
This is due largely to the fact that the steam-ports are so small 
that the steam cannot get into the cjdinder fast enough to exert 
its full pressure. It is often purposely wire-drawn by partially 
closing the throttle, so that the steam may be used less rapidly. 

(b) Entrained water. Steam is always drawn from a dome 
placed over the boiler so that the steam shall be as far above 
the water-surface as possible, and shall be as dry as possible. 
In spite of this the steam is not perfectly dry and carries with 
it water at a temperature of, say, 361°, and pressure of 140 lbs. 
per square inch. When the pressure falls during the expan- 
sion and exhaust, this hot water turns into steam and absorbs 
the necessary heat from the hot cylinder-walls. This heat is 
then carried out by the exhaust and wasted. 

(c) The back pressure of the exhaust-steam, which depends 
on the form of the exhaust-passages, etc. This amounts to 
from 2 to 20% of the power developed. 

(d) Clearance-spaces. When cutting off at full stroke this 
waste is considerable (7 to 9%), but when the steam is used 
expansively the steam in these clearance-spaces expands and 
so its power is not wholly lost. 

(c) Radiation. In spite of all possible care in jacketing the 
cylinders, some heat is lost by radiation. 

(/) Radiation into the exhaust-steam. This is somewhat 
analogous to (&). Steam enters the C3dinder at a temperature 
of, say, 361°; the walls of the cylinder are much cooler, say 250°; 
some heat is used in raising the temperature of the cylinder- 
walls; some steam is vaporized in so doing; when the exhaust 
is opened the temperature and pressure fall; the heat tem- 
porarily absorbed by the cylinder-walls is reabsorbed by the 
exhaust-steam, re-evaporating the vapor pre\dously formed, 
and thus a certain portion of heat-energy goes through the 
cylinder vathout doing any useful \\'ork. With an early cut-off 
the loss due to this cause is very great. 

The sum of all these losses is exceedingly variable. They 
are usually less at lower speeds. The loss in initial pressure 



•378 RAILROAD CONSTRUCTION. § 322. 

(the difference between boiler pressure and the cylinder pres- 
sure at the beginning of the stroke) is frequently over 20%, 
but this is not all a net loss With an early cut-off the average 
cylinder pressure for the whole stroke is but a small part of 
the boiler pressure, yet the horse -power developed may be as 
great as, or greater than^ that developed at a lower speed, later 
cut-off; and higher average pressure- 

322. Tractive power The work done by the two cylinders 
during a complete revolution of the drivers evidentl}^ =area of 
pistons X average steam pressure X stroke X 2X2. The resist- 
ance overcome evidently = tractive force at circumference of 
drivers times distance traveled by drivers (which is the cir- 
cumference of the drivers) Therefore 

C area pistons X average steam pressure 

rr. .- r < XstrokeX2X2, 

Tractive force = ) -. ^ „-,—. — . 

C circumference of drivers 

Dividing numerator and denominator by tz (3.1415), we have 

C (diam piston) ^ x average steam 

Tractive forces j. P'-e^^^ X stroke _ 

( diameter of driver 

which is the usual rule Although the rule is generally stated 
in this form, there are several deductions In the first place 
the net effective area of the piston is less than the nominal on 
account of the area of the piston-rod. The ratio of the areas 
of the piston-rod and piston varies, but the effect of this reduc- 
tion is usually from 1.3 to 1-7%. No allowance has been made 
for friction — of the piston, piston-rod, cross-head, and the 
various bearings This would make a still further reduction 
of several per cent. Nevertheless the above simple rule is 
used, because^ as will be shown, no great accuracy can be 
utilized. 

The tractive force is limited by the adhesion between the 
drivers and the rails, and this is a function of the weight on the 
drivers. Under the most favorable conditions this has been 
tested to amount to one-third the weight on the drivers, but 
such a ratio cannot be depended on Wellington used the 
ratio one-fourth The Baldwin Locomotive Works in their 
•* Locomotive Data" give tables and diagrams based on J, /^, 



§ 322. KOLLING-STOCK. 379 

and i adhesion. As low a value as ^ or even ^ is occasionally 
used, but such a low rate of adhesion would only be found when 
the rails were abnormally slippery. In a well-designed loco- 
motive the tractive force, as computed above, and the tractive 
adhesion should be made about equal. The uncertainty in 
the coefficient of adhesion shows the futility of any refinement 
in the computation of the tractive force. 

It is only at very slow speeds that an engine can utilize all 
of its tractive force. When running at a high speed, the utmost 
horse-power that the engine can develop will only produce a 
draw-bar pull, which is but a small part of the possible tractive 
force. Power is the product of force times velocity. If the 
power is constant and the velocity increases, the force must 
decrease. This fact is well shown in the figures of some tests 
of a locomotive. The dimensions were as follows: cylinders, 
18''X24''; drivers, 68"; weight on drivers, 60000 lbs. ; heating- 
surface, 1458 sq. ft.; grate-area, 17 sq. ft. During one test 
the average cylinder pressure was 83.3 lbs. (boiler pressure, 
145; 14-inch . cut-off and throttle f open). By the above 
formula (137), 

Tractive force = xi = 9525 lbs. 

— bo 

At i adhesion the tractive force was 15000 lbs; even at J ad- 
hesion, it would be 12000 lbs. This shows that at the speed 
of :his test (26.3 in. per hour) scarcely more than f of the trac- 
tive power w^as utilized. A still more marked case, shown by 
another test with the same engine, taken when the speed was 
53.4 miles per hour, indicated an average cylinder pressure of 
37.2 lbs., the throttle being } open and the vah^es cutting off 
at 8". In this case the tractive power, computed as before, 
equals 4254 lbs., about ^^ of the weight on the drivers and 
about i of the tractive force which is possible at slow speeds. 
In the first case, the tractive power (9525) times the speed in 
feet per second (38.57) divided by 550 gives the indicated horse- 
power, 668. In the second case, although the tractive force 
developed was so miich less, the speed was much greater and 
the horse-power was about the same, 606. 

The above figures illustrate some of the foregoing statements 
regarding loss of efficiency. In both cases the steam was wire- 
drawn. The boiler pressure was 145 lbs., but when the throttle 



380 RAILROAD CONSTRUCTION. § 322. 

was only f open and the steam was cut-ofT at 14'' (24" stroke) 
the average steam pressure in the cylinder was reduced to 
83.3 lbs. With the throttle but i open and the valves cutting 
off at 8" (i of the stroke), the average pressure was cut down 
to 37.2 lbs. — about J of the boiler pressure. Note that the heat- 
ing-surface per square foot of grate-area (1458-^17 = 86) is 
very large (see § 320) . Note also that the horse-power developed 
divided by the grate-area (17) gives 39 and 36 H.P. per square 
foot of grate-area. This is exceptionally large — 25 or 30 being 
a more common figure. 

The maximum tractive power is required when a train is 
starting, and fortunately it is at low velocities that the maxi- 
mum tractive force can be developed. The motion of the 
piston is so slow that there is but little reduction of steam 
pressure, and the valves are generally placed to cut off at full 
stroke. For the above engine, with 145 lbs. boiler pressure, 

18^X145X24 

the absolute maximum of tractive force is ^^ = 

bo 

16581 lbs. Of course, this maximum would never be reached 
unless the boiler pressure were increased. A common rule is 
to consider that the average effective cylinder pressure for slow 
speed and full stroke will be 80% of the boiler pressure. This 
would reduce the tractive force to the (nominal) value of 13265 
lbs., and the corresponding cylinder pressure would be 116 lbs. 
per square inch. With an effective cylinder pressure of about 
131 lbs. the tractive power is 15000 lbs., which is J of the total 
weight on the drivers. This illustrates the general rule, stated 
above, that the cylinders, drivers, and boiler pressure should 
be so proportioned that the maximum tractive force should 
about equal the maximum adhesion which could be obtained. 

As another numerical example, the dimensions of a recently 
constructed heavy consolidation engine are quoted. The cylin- 
ders are 24''X32''; diameter of drivers, 54''; total weight of 
engine and tender, 391400 lbs.; weight of engine, 250300 lbs.; 
weight on drivers, 225200 lbs.; capacity of tender, 7500 gallons; 
the boiler has 406 tubes, 21" in diameter and 15' long; fire- 
box, 132"X401-"; heating-surface of tubes, 3564 sq. ft.; of 
lire-box, 241 sq. ft. — total, 3805 sq. ft.; boiler pressure, 220 lbs. 
per square inch. Applying Eq. 132, we may compute 75093 
lbs. as the absolute maximum of tractive power. In fact this 
is an unattainable limit, for reasons before stated, The trac* 



§ 323. ROLLING-STOCK. SSI 

tive force is given as 63000, which corresponds to an effective 
cyhnder pressure of about 185 lbs., about 84% of the boiler 
pressure. This tractive force is 28% of the weight on the 
drivers, a tractive ratio of 1 : 3.6. 

RUNNING GEAR. 

323. Types of running gear, (a) "American." This was 

Q/^~\ once the almost universal type for 
^^-^ LJ L2 -^ both passenger and freight service. 

It is still very commonly used for passenger service, but it is 
not the best form for heavy freight work. 

(b) "Columbia." Four drivers, one pair of pilot-truck wheels 
and one pair of trailing wheels be- /^^ /^ ^^ 

hind the drivers. The low trailing ^ \ ■/ lk_^ — ^ -^ 

wheels permit a desirable enlargement of the fire-box. This 
is a recent type, used exclusively for passenger service. 

Q^^ (c) "Atlantic." Similar to 
\ J O Q -^ h except that the pilot-truck 

has four wheels instead of two. 

(d) "Mogul." These are used for both passenger and freight 
service, but are not well 
adapted for either high speedT 
or great tractive power. 

(e) "Ten-wheel." Similar to d except that the pilot-truck 

Q^->^ •^ has four wheels instead of 

\ J V J 00 -^ two. The use is similar to 
that of d. 

(f) "Consolidation." The present standard for freight ser- 
vice. It permits great trac- /^^ /^^ /^~^ /^'^ 

tive power without excessive \. y ^y \ x \ y LJ -^ 

concentrated loads on the track. 

(g) Switching-engines. These have four or six (and excep- 
tionally even eight or ten) drivers and no truck-wheels. They 
are only adapted for slow speed when a maximum of tractive 
power is needed. Sometimes the water-tank and even a small 
fuel-box is loaded on. Since fuel is always near at hand for a 
3^ard -engine, the fuel-box need not be large. 

(h) "Double-enders." As explained in § 315, truck-wheels are 
needed in front of the drivers to guide them around curves. If 
an ordinary engine is run backward, the flanges of the rear 



o O O o - 



382 RAILROAD CONSTRUCTION. § 324. 

drivers will become badly worn, and if the speed is high, the 

danger of derailment is considerable. In suburban service, 

-^^ /^ /^^ ^^ when the runs are short, it is 

sZ — )^i^ ^^-^ — ^ preferable to run the engines 

forward and backward, rather than turn them at each end of 
the route. Therefore a pilot-truck is placed at each end. 

(i) "Miscellaneous types." Almost every conceivable com- 
bination of drivers and truck-wheels has been used. The 
^' Mastodon'^ is similar to the '^Consolidation'' except that the 
pilot-truck has four wheels instead of two. The "Decapod" 
has ten driving-wheels. The '^Forney" (named after the in- 
ventor) has been very extensively used on elevated roads. The 
weight of the boiler and machinery is carried on four driving- 
wheels; the engine-frame is extended so as to include a small 
tank and fuel-box, the weight of which is chiefly supported by 
a truck of two or four wheels. They run best when running 
"backward," i.e., tender first. 

The great variation in types of running gear which has been 
developed in recent years, has started a convenient and unmis- 
takable method of indicating the running gear. Commencing 
with the front of the engine (or pilot) always at the left (instead 
of at the right, as in the illustrations above) the number of 
wheels of the pilot truck on both rails is indicated by 0, 2 or 4, 
according as there is no pilot truck, a two- wheeled or a four- 
wheeled pilot truck. Then the number of drivers on both rails 
is indicated by the next number and the number of trailing 
wheels by the third number. The running gear of the tender is 
not indicated. This method may be illustrated by applying it 
to the types indicated above: 

American 4-4-0 

Columbia 2-4-2 

Atlantic 4-4-2 

Mogul 2-6-0 

Ten-wheel . 4-6-0 

Consolidation 2-8-0 

Six- wheel switcher 0-6-0 

Mastodon 4-8-0 

The running gear of any new type may thus be unmistakably 
indicated by three figures. 



§ 324. ROLLING-STOCK. 383 

324. Equalizing-bvars. The ideal condition of track, from 
the standpoint of smooth running of the roUing stock, is that 
the rails should always lie in a plane surface. While this con- 
dition is theoretically possible on tangents, it is unobtainable 
on curves, and especially on the approaches to curves when the 
outer rail is being raised. Even on tangents it is impossible 
to maintain a perfect surface, no matter how perfectly the 
track may have been laid. In consequence of this, the points 
of contact of the wheels of a locomotive, or even of a four- 
wheeled truck, will not ordinarily lie in one plane. The rougher 
and more defective the track, the worse the condition in th's 
respect. Since the frame of a locomotive is practically rigio^^ 
if the frame rests on the driver-axles through the medium of 
springs at each axle-bearing, the compression of the springs 
(and hence the pressure of the drivers on the rail) will be varia- 
ble if the bearing-points of the drivers are not in one plane 
This variable pressure affects the tractive power and severely 
strains the frame. Applying the principle that a tripod will 
stand on an uneven surface, a mechanism is employed which 
virtually supports the locomotive on three points, of which one 
is usually the center-bearing of the for^v^ard truck. On each 
side the pressure is so distributed among the drivers that even 
if a driver rises or falls with_reference to the others, the load 
carried by each driver is unaltered, and that side of the engine 
rises or falls by one nth of the rise or fall of the single driver, 
where n represents the number of wheels. The principle in- 
volved is shown in an exaggerated form in Fig. 195. In the 
diagram, MN represents the normal position of the frame when 
the wheels are on line. The frame is supported by the hangers 
at a, c, /, and h. ah, de, and gh are horizontal levers vibrating 
about the points H, K, and L, which are supported by the 
axles. While it is possible with such a system of levers to make 
MN assume a position not parallel with its natural position, 
yet, by an extension of the principle that a beam balance loaded 
with equal weights will always be horizontal, the effect of rais- 
ing or lowering a wheel will be to move ikfiV parallel to itself. 
It only remains to determine how much is the motion of MN 
relative to the rise or drop of the wheel. 

The dotted lines represent the positions of the wheels and 
levers when one wheel drops into a depression. The wheel 
center drops from p to g, a distance m, L drops to U, a 
distance m (see Fig. 195, h) ; M drops to M', an unknown dis- 



384 



RAILROAD CONSTRUCTION. 



§324. 



tance x; therefore aa^=x; hV =x] cc^=x] dd^=Sx = ee'; ff =x) 
.'.gg^ = 5x; hh^=x; LU = i(gg^ + hh^) =^(6x)=m; .\x = lm; 
i.e., MN drops, parallel to itself, 1/n as much as the wheel 
drops, where n is the number of wheels. The resultant effect 
caused by the simultaneous motion oF two wheels with refer- 

H -/ d K a ^:. L 




Fig. 195. — Action of Equalizing-levers. 
ence to the third is evidently the algebraic sum of the effects 
of each wheel taken separately. 

The practical benefits of this device are therefore as follows : 

(a) When any driver reaches a rough place in the track, a 
high place or a low place, the stress in all the various hangers 
and levers is unchanged. 

(h) The motion of the frame (represented by the bar MN 
in Fig. 195) is but 1/n of the motion of the wheel, and the jar 
and vibration caused by a roughness in the track is correspond- 
ingly reduced. 

The details of applying these principles are varied, but in 
general it is done as follows; 

(a) American and ten-wheeled types. Drivers on each side 
form a system. The center-bearing pilot-truck, is the third 
point of support. The method is illustrated in Fig. 196. 

(b) Mogul and consolidation types. The front pair of drivers 
is connected with the two-wheeled pilot-truck (as illustrated 
in Fig. 197) to form one system. The remaining drivers on 
each side are each formed into a system 

The device of equalizers is an American invention. Until 
recently it has not been used on foreign locomotives. The 
necessity for its use becomes less as the track is maintained 



I 



§ 324. 



ROLLING-STOCK. 



385 



with greater perfection and is more free from sharp curves. 
A locomotive not equipped with this device would deteriorate 





very rapidly on the comparatively rough tracks which are 
usually found on light-traffic roads. It is still an open ques- 



386 EAILROAD CONSTRUCTION. § 325. 

tion to what extent the neglect of this device is responsible 
for the statistical fact that average freight-train loads on foreign 
trains are less in proportion to the weight on the drivers than 
is the cas^ with American practice. The recent increasing use 
of this device on foreign heavy freight locomotives is perhaps 
an acknowledgment of this principle. 

325. Counterbalancing. At very high velocities the cen- 
trifugal force developed by the weight of the rotating parts 
becomes a quantity which cannot be safely neglected. These 
rotating parts include the crank-pin, the crank-pin boss, the 
side rod, and that part of the weight of the connecting-rod 
which may be considered as rotating about the center of the 
crank-driver. As a numerical illustration, a driving-wheel 
62'' in diameter, running 60 miles per hour, will revolve 325 
times per minute. The weights are: 

Crank-pin 110 lbs. 

boss 150 '' 

One-half side rod 240 '' 

Back end of connecting-rod. ........ 190 * ' 

Total 690 lbs. 

If the stroke is 24", the radius of rotation is 12", or 1 foot. Then 
Gv' 690X471^^X3252 ^^^^^ ., 
^= 32.2X1X6Q2 =24821 lbs., 

which is half as much again as the weight on a driver, 16000 lbs. 
Therefore if no counterbalancing were used, the pressure be- 
tween the drivers and the rail would always be less (at any 
velocity) when the crank-pin was at its highest point. At a 
velocity of about 48 miles per hour the pressure would become 
zero, and at higher velocities the wheel would actually be 
thrown from the rail. As an additional objection, when the 
crank-pin was at the lowest point, the rail pressure would be 
increased (velocity 60 miles per hour) from 16000 lbs. to nearly 
41000 lbs., an objectionably high pressure. These injurious 
effects are neutralized by "counterbalancing.'' Since all of 
the above-mentioned weights can be considered as concen- 
trated at the center of the crank-pin, if a sufficient weight is so 
placed in the drivers that the center of gravity of the eccentric 
weight is diametrically opposite to the crank-pin, this centrifu- 
gal force can be wholly balanced. This is done by filling up 
a portion of the space between the spokes. If the center of 
gravity of the counterbalancing weight is 20" from the center. 



§ 325. ROLLING STOCK. 387 

then, since the crank-pin radius *s 12", the required weight 
would be 690x11 = 414 lbs. 

In addition to the effect of these revolving parts there is 
the effect of the sudden acceleration and retardation of the 
reciprocating parts. In the engine above considered the weights 
of these reciprocating parts will be: 

Front end of connecting-rod 150 lbs. 

Cross-head 174 ^ ^ 

Piston and piston-rod 300 '' 

Total 624 lbs. 

Assume as before that the reciprocating parts may be con- 
sidered as concentrated at one point, the point P of the dia- 
gram in Fig. 198. Since the 

motion of P is horizontal ,^''^ " ""^W' 

only, the force required to / /^TTTT^' \ 

•+ • ^- + / / ! >^W \ 
overcome its mertia at any /^_^ \ I J^_Li__l_. 

point will exactly equal p^ ■ — ^"■^-A.^^VL/i^ / '' 

the horizontal component of v^"^^^^!!.^^^ / 

the force required to over- ^^-^^ ^X 

come the inertia of an equal 

weight at S revolving in ^^^- 198.-Action of Colxtekbalance. 

a circular path. Then evidently the horizontal component of 
the force required to keep TT^ in the circular path will exactly 
balance the force required to overcome the inertia of P. Of 
course W=P, But a smaller weight TT^', w^hose weight is 
inversely proportional to its radius of rotation, wdll e^ddently 
accompHsh the same result. In the above nimaerical case, if 
the center of gravity of the counterweights is 20" from the 
center, the required weight to completely counterbalance 
the reciprocating parts w^ould be 624 X if = 374.4 lbs. This 
counterweight need not be all placed on the driver carr^^ing 
the main crank-pin, but can be (and is) distributed among all 
the drivers. Suppose it were divided between the two drivers 
in the above case. At 60 miles per hour such a counterweight 
w^ould produce an additional pressure of 11211 lbs. when the 
counterw^eight w^as down, or a lifting force of the same amount 
when the counterw^eight was up. Although this is not suffi- 
cient to lift the driver from the rail, it would produce an objec- 
tionably high pressure on the rail (over 27000 lbs.), thus inducing 
just what it was desired to avoid on account of the eccentric 
rotating parts. Therefore a compromise must be made. Only 
a portion (one half to three fourths) of the weight of the recip- 
rocating parts is balanced. Since the effect of the rotating 



388 RAILROAD CONSTRUCTION. § 325. 

weights is to cause variable pressure on the rail, while the effect 
of the reciprocating parts is to cause a horizontal wobbling or 
*^ nosing'' of the locomotive, it is impossible to balance both. 
Enough counterweight is introduced to partially neutralize the 
effect of the reciprocating parts, still leaving some tendency 
to horizontal wobbling, while the counterweights which were 
introduced to reduce the wobbling cause some variation of 
pressure. By using hollow piston-rods of steel, ribbed cross- 
heads, and connecting- and side-rods with an I section, the 
weight of the reciprocating parts may be greatly lessened with- 
out reducing their strength, and with a decrease in weight the 
effect of the unbalanced reciprocating parts and of the " excess 
balance'' (that used to balance the reciprocating parts) is 
largely reduced. 

Current practice is somewhat variable on three features: 

(a) The proportion of the weight of the connecting-rod which 
should be considered as revolving weight. 

(b) The proportion of the total reciprocating weight that 
should be balanced. 

(c) The distribution among the drivers of the counterweight 
to balance the reciprocating parts. 

An exact theoretical analysis of (a) shows that it is a func- 
tion of the weights and dimensions of the reciprocating parts. 
The weight which may be considered as revolving equals * 






i) 



in which r= radius of the crank, Z= length of connecting-rod, 
A: = distance of center of gyration from wrist-pin, <i= distance 
of center of gravity from wrist-pin, Tri= weight of connecting- 
rod in pounds, and 1^2= weight of piston, piston-rod, and cross- 
head in pounds; all dimensions in feet. An application of this 
formula will show that for the dimensions of usual practice, 
from 51 to 57% of the weight of the connecting-rod should be 
considered as revolving weight. 

The principal rules which have been formulated for counter- 
balancing may be stated as follows: 

1. Each wheel should be balanced correctly for the revolving 
parts connected with it. 

2. In addition, introduce counterbalance sufficient for 50% 
of the weight of the reciprocating parts for ordinary engines, 

* R. A. Parke, in R. R. Gazette, Feb. 23, 1894. 



326. 



ROLLING-STOCK. 



389 



increasing this to 75% when the reciprocating parts are exces- 
sively heavy (as in compound locomotives) or when the engine 
is Hght and unable to withstand much lateral strain or when 
the wheel-base is short. 

3. Consider the weight of the connecting-rod as J revolving 
and I reciprocating when it is over 8 feet long; when shorter 
than 8 feet, consider ^^ of the weight as revolving and x% as 
reciprocating. 

4. The part of the weight of the connecting-rod considered 
as revolving should be entirely balanced in the crank-driver wheel 

5. The '^excess balance" should be divided equally among 
the drivers. 

6. Place the counterbalance as near the rim of the wheel 
as possible and also as near the outside 
of the wheel as possible in order that 
the center of gravity shall be as near 
as possible opposite the center of 
gravity of the rods, etc., which are all 
outside of even the plane of the face 
of the wheel. 

In Fig. 199 is shown a section of a 
locomotive driver with the cavities in 
the casting for the accommodation of 
the lead which is used for the counter- 
balance weight. Incidentally several 
other features and dimensions are shown 
in the illustration. 

326. Mutual relations of the boiler power, tractive power, 
and cylinder power for various types. The design of a locomo- 
tive includes three distinct features w^hich are varied in their 
mutual relations according to the work w^hich the engine is 
expected to do. 

(a) The boiler power. This is limited by the rate at which 
steam may be generated in a boiler of admissible size and weight. 
Engines which are designed to haul very fast trains which are 
comparatively light must be equipped with very large grates and 
heating surfaces so that steam may be developed with great 
rapidity in order to keep up with the very rapid consumption. 
Engines for very heavy freight work are run at very much 
lower velocity and at a lower piston speed in spite of the fact 
that more strokes are required to cover a given distance and 
the demand on the boiler for ra'pid steam production is not 




Fig, 199. — Section of 
Locomotive-driver. 



390 RAILROAD CONSTRUCTION. § 326* 

as great as with high-speed passenger-engines. The capacity of 
a boiler to produce steam is therefore limited by the limiting 
weight of the general type of engine required. Although im- 
provements may be and have been made in the design of fire- 
boxes so as to increase the steam-producing capacity without 
adding proportionately to the weight, yet there is a more or 
less definite limit to the boiler power of an engine of given 
weight. 

(b) The tractive power. This is a function of the weight on 
the drivers. The absolute limit of tractive adhesion between a 
steel-tired wheel and a steel rail is about one third of the pressure, 
but not more than one fourth of the weight on the drivers can 
be depended on for adhesion and wet rails will often reduce 
this to one fifth and even less. The tractive power is therefore 
absolutely limited by the practicable weight of the engine. In 
some designs, when the maximum tractive power is desired, not 
only is the entire weight of the boiler and running gear thrown 
on the drivers, but even the tank and fuel-box are loaded on. 
Such designs are generally employed in switching-engines (or 
on engines designed for use on abnormally heavy mountain 
grades) in which the maximum tractive power is required, but 
in which there is no great tax on the boiler for rajpid steam pro- 
duction (the speed being always very low), and the boiler and 
fire-box, which furnish the great bulk of the weight of an engine, 
are therefore comparatively light, and the requisite weight for 
traction must, therefore, be obtained b}- loading the drivers 
as much as possible. On the other hand, engines of the highest 
speed cannot possibly produce steam fast enough to maintain 
the required speed unless the load be cut do^\Ti to a compara- 
tively small amount. The tractive power required for this 
comparatively small load will be but a small part of the weight 
of the engine, and therefore engines of this class have but a 
small proportion of their weight on the drivers; generally 
have but two driving-axles and sometimes but one. 

(c) Cylinder power. The running gear forms a mechanism 
which is simply a means of transforming the energy of the boiler 
into tractive force and its power is unlimited, within the prac- 
tical conditions of the problem. The power of the running 
gear depends on the steam pressure, on the area of the piston, 
on the diameter of the drivers, and on the ratio of crank-pin 
radius to wheel radius, or of stroke to driver diameter. It 



§326. 



ROLLING-STOCK. 



391 



is always possible to increase one or more of these elements 
by a relatively small increase of expenditure until the cylinders 
are able to make the drivers slip, assuming a sufficiently great 
resistance. Since the power of the engine is limited by the 
power of its weakest feature, and since the running gear is the 
most easily controlled feature, the power of the running gear 
(or the ^^ cylinder power") is always made somewhat excessive 
on all well-designed engines. It indicates a badly designed 
engine if it is stalled and unable to move its drivers, the steam 
pressure being normal. If it is attempted to use a freight- 
engine on fast passenger service, it will probably fail to attain 
the desired speed on account of the steam pressure falling. 
The tractive power and cyhnder power are superabundant, but 
the boiler cannot make steam as fast as it is needed for high 
speed, especially when the drivers are small. The practical 
result would be a comparatively low speed kept up with a forced 
fire. If it is attempted to use a high-speed passenger-engine 
on heavy freight ser\dce, the logical result is a slipping of the 
drivers until the load is reduced. The boiler power and cylinder 
power are ample, but the weight on the drivers is so small that 
the tractive power is only sufficient to draw a comparatively 
small load. 

These relations between boiler, cylinder, and tractive power 
are illustrated in the following comparative figures referring 
to a fast passenger-engine, a heavv freight-engine, and a switch- 
ing-engine. The weights of the passenger- and freight-engines 
are about the same, but the passenger-engine has only 72% of 





Cylinders. 


Total 
Wght. 


Wt. on 
Driv'rs 


Heat- 
ing 
Sur- 
face, 

sq. ft. 


Grate 
area 

sq. ft. 


Steam 
Pres- 
sure in 
Boiler. 


Stroke. 


Kind. 


Diam. 
Driver. 


Fast passenger . 
Heavy freight . 
Switcher 


19''X24" 

20''X24" 
19''X24'' 


126700 
128700 
109000 


81500 
112600 
109000 


1831.8 
1498.3 
1498.0 


26.2 
31.5 

22.8 


180 
140 
160 





the tractive power of the freight. But the passenger-engine 
has 22% more heating-surface and can generate steam much 
faster; it makes less than two thirds as many strokes in cover- 
ing a given distance, but it runs at perhaps twice the speed 



392 RAILROAD CONSTRUCTION. § 326, 

and probably consumes steam much faster. The switch- 
engine is lighter in total weight, but the tractive power is nearly 
as great as the freight and much greater than the passenger- 
engine. While the heating-surfaces of the freight- and switch- 
ing-engines are practically identical, the grate area of the switcher 
is much less; its speed is always low and there is but little neces- 
sity for rapid steam development. 

While these figures show the general tendency for the relative 
proportions, and in this respect may be considered as typical, 
there are large variations. The recent enormous increase in 
the dead weight of passenger-trains has necessitated greater 
tractive power. This has been provided sometimes by using 
''Mogul" and 'Hen-wheel'' engines, which were originally 
designed for freight work. On the other hand, the demand 
for fast freight service, and the possibility of safely operating 
such trains by the use of air-brakes, has required that heavy 
freight-engines shall be run at comparatively high speeds, and 
that requires the rapid production of steam, large grate areas, 
and heating surfaces. But in spite of these variations, the 
normal standard for passenger service is a four-driver engine 
carrying about two thirds of the weight of the engine on the 
drivers, which are very large; the normal standard for freight 
work is the ''consolidation," with perhaps 90% of the weight 
on the drivers, which are small, but which must have the pony 
truck for such speed as it uses; and finally the normal standard 
for switching service has all the w^eight on the drivers and has 
comparatively low stea]Ti-producing capacity. 

327. Life of locomotives. The life of locomotives (as a 
whole) may be taken as about 800000 miles or about 22 to 24 
years. While its life should be and is considered as the period 
between its construction and its final consignment to the scrap 
pile, parts of the locomotive may have been renewed more 
than once. The boiler and fire-box are especially subject to 
renew^al. The mileage life is much longer than formerl3^ This 
is due partly to better design and partly to the custom of 
drawing the fires less frequently and thereby avoiding some 
of the destructive strains caused by extreme alternations of 
heat and cold. Recent statistics give the average annual 
mileage on twenty-three leading roads to be 41000 miles. 



§ 328. ROLLING-STOCK. 393 

CARS. 

328. Capacity and size of cars. The capacity of freight-cars 
has been enormously increased of late years. About thirty 
years ago the usual live-load capacity for a box-car was about 
20000 lbs. In 1893 the standard box-car, gondola-cars, etc., 
of the Pennsylvania Railroad on exhibition at the Chicago 
Exposition, had a live-load capacity of 60000 lbs. and a dead 
weight of 30000 to 33000 lbs With a full load, the weight on 
each wheel is nearly 12000 lbs , which equals or exceeds the 
load usually placed on the drivers of ordinary locomotives. 
But now cars with a hve-load capacity of 100,000 lbs. are stand- 
ard on many roads, and even larger cars are made for special 
service. (See Fig. 200.) 

The limitation of the carr\^ing capacity for some kinds of 
freight depends somewhat on the amount of live load that 
can be carried wdthin given dimensions; for the cross-section of 
a car is limited to the extreme dimensions which may be safely 
run through the tunnels and through bridges as at present 
constructed, and the length is somewhat Hmited by the dif- 
ficulty of properly supporting an excessively heavy load, dis- 
tributed over an unusually long span, by a structure which is 
subjected to excessive jar, concussion, compression, and ten- 
sion. The cross-sectional limit seems to have been scarcely 
reached yet, except, perhaps, in the case of furniture and carriage- 
cars, whose load per cubic foot is not great. The usual width 
of freight-cars is about 8 to 9 feet, w^hile parlor-cars and sleepers 
are generally 10 feet wdde and sometimes 11 feet. The highest 
point of a train is usually the smoke-stack of the locomotive 
which is generally 14 feet above the rails and occasionally over 
15 feet. A sleeping-car usually has the highest point of the 
car about 14 feet above the rails. Box-cars are usually about 
8 feet high (above the sills), with a total height of about 11' 3". 
Refrigerator-cars are usually about 9' high and furniture-cars 
about 10' above the sills, the truck adding about 3' 3". The 
usual length is 34 feet, but 35 to 40 feet is not uncommon. 
Passenger-cars (day coaches) are usually 50 feet long, exclusive of 
the end platforms and weigh 45000 to 50000 lbs. Sixty pas- 
sengers at 150 pounds apiece (a high average) will only add 
9000 lbs. to the weight. A parlor-car or sleeper is generally 
about 65 feet long exclusi\'e of the platforms, which add about 
6' 6". The weight is anywhere from 60000 to 80000 lbs- 



394 RAILROAD CONSTRtJCTION. § 329. 

'The weight of the 25 or 30 passengers it may carry is hardly 
worth considering in comparison. 

329. Stresses to which car-frames are subjected. A car 
is structurally a truss, supported at points at some distance 
from the ends and subjected to transverse stress. There is, 
therefore, a change of flexure at two points between the trucks. 
Besides this stress the floor is subjected to compression when 
the cars are suddenly stopped and to tension when in ordinary 
motion, the tension being greater as the train resistance is 
geater and as the car is nearer the engine. The shocks, jars, 
and sudden strains to which the car-frames are subjected are 
very much harder on them than the mere static strains due to 
their maximum loads if the loads were quiescent. Consequently 
any calculations based on the static loads are practically value- 
less, except as a very rough guide, and previous experience 
must be relied on in designing car bodies. As evidence of the 
increasing demand for strength in car-frames, it has been re- 
cently observed that freight-cars, built some years ago and 
built almost entirely of wood, are requiring repairs of wooden 
parts which have been crushed in service, the wood being per- 
fectly sound as regards decay. 

330. The use of metal. The use of metal in car construction 
is very rapidly increasing.. The demand for greater strength 
in car-frames has grown until the wooden framing has become 
so heavy that it is found possible to make steel frames and 
trucks at a small additional cost, the steel frames being twice 
as strong and yet reducing the dead weight of the car about 
5000 lbs., a consideration of no small value, especially on roads 
having heavy grades. Another reason for the increasing use 
bf metal is the great reduction in the price of rolled or pressed 
steel, while the cost of wood is possibly higher than before. 
The advocates of the use of steel advise steel floors, sides, etc. 
For box-cars a wooden floor has advantages. For ore and 
coal-cars an all-metal construction has advantages. (Fig. 200.) 
In Germany, where steel frames have been almost exclusively 
in use for many years, they have not 3^et been able to determine 
the normal age limit of such frames; none have yet worn onto 
The life is estimated at 50 to 80 years 



ii«|gi 



:100,000-LB. Box Car. 




Steel Coal Car. 




Wooden Box Car; Steel Framb.s 
Fig. 200. — Some Heavy Freight GabM» 
(To face page 394.) 



§ 331. EOLLING-STOCK* 395 

Brake -beams are also best made of metal rather than wood, 
as was formerly done. Metal brake-beams are generally used on 
cars having air-brakes, as a wooden beam must be excessively 
large and heavy in order to have sufficient rigidity. 

Truck-frames (see Fig. 201), which were formerly made prin- 
cipally of wood, are now largely made of pressed steel. It makes 




Fig 2Q1. 

a reduction in weight of about 3000 lbs. per car. The increased 
durability is still an uncertain quantity. 

331. Draft gear. The enormous increase in the weight and 
live load capacities of rolling stock have necessitated a corre- 
sponding development in draft gear. Even within recent years, 
'^ coal- jimmies/' carrying a few tons have been made up into 
trains by dropping a chain of three big links over hooks on the 
ends of the cars. But the great stresses due to present loadings 
would tear such hooks from the cars or tear the cars apart if 
such cars were used in the make-up of long heavy trains as now 
operated. The next stage in the development of draft gear was 
the invention of the ^^ spring coupler," by which the energy due 
to a sudden tensile jerk or the impact of compression may be 
absorbed by heavy springs and gradually imparted to the car 
body. Such devices, for which there are many designs, seemed 
to answer the purpose for cars of 25 to 40 tons capacity. The 
use of 100,000-pound steel cars soon proved the inadequacy of 
even spring couplers. The friction-draft gear was then in- 
vented. The general principle of such a gear is that, when 



396 



EAiLItdAD CONSTEUCTION, 




§331. 




§ 332. ROLLING-STOCK. 397 

acting at or near its maximum capacity, it harmlessly trans- 
forms into heat the excessive energy developed by jerks or 
compression. There are several different designs of such gear, 
but the general principle underlying all of them may be illus- 
trated by a description of the Westinghouse draft gear. The 
gear employs springs which have sufficient stiffness to act as 
ordinary spring-couplers for the ordinary pushing and pulling 
of train operations. Sections of the gear are shown in Fig. 202, 
while the method of its application to the framing of a car of 
the pressed steel type is shown in Fig. 203, a and &. When 
the draft gear is in tension the coupler, which is rigidly attached 
to By is drawn to the left, drawing the follower Z with it. Com- 
pression is then exerted through the gear mechanism to the 
follower A which, being restrained by the shoulders RR, against 
which it presses, causes the gear to absorb the compression. 
The coil-spring C forces the eight wedges n against the eight 
corresponding segments E, The great compression of these 
surfaces against the outer shell produces a friction which retards 
the compression of the gear. The total possible movement of 
the gear, as determined by an official test, was 2.42 inches, when 
the maximum stress v/as 180,000 pounds. The work done in 
producing this stress amoimted to 18,399 foot-pounds. Of this 
total energy 16,666 foot-pounds, or over 90%, represents the 
amount of energy absorbed and dissipated as heat by the 
frictional gear. The remaining 10% is given back by the 
recoil. The main release spring K is used for returning the 
segments and friction strips to their normal position after the 
force to close them has been removed. It also gives additional 
capacity to the entire mechanism. The auxiliary spring L 
releases the wedge Z), while the release pin M releases the pres- 
sure of the auxiliary spring L against the wedge during fric- 
tional operation. If we omit from the above design the fric- 
tional features and consider only the two followers A and Z, 
separated by the springs C and X, acting as one spring, we have 
the essential elements of a spring-draft gear. In fact, this 
gear acts exactly like a spring-draft gear for all ordinary service, 
the frictional device only acting during severe tension and com- 
pression. 

332. Gauge of wheels and form of wheel-tread. — In Fig. 204 
is shown the standard adopted by the Master Car Builders' 
^Association at their twentieth annual convention. Note the 



398 



RAILROAD CONSTRUCTION. 



§ 332. 



o o o 



o o o 



k . ^ ^^ ^ 





§ 333. ROLLING-STOCK. 399 

normal position of the gauge-line on the wheel-tread. In 
Fig. 114, p. 259, the relation of rail to wheel-tread is showTi 
on a smaller scale. It should be noted that there is no definite 
position where the wheel-flange is absolutely "chock-a-block" 
against the rail. As the pressure increases the wheel mounts 
a little higher on the rail until a point is soon reached when the 
resistance is too great for it to mount still higher. By this 
means is avoided the shock of unyielding impact when the car 
sways from side to side. When the gauge between the inner 
faces of the wheels is greater or less than the limits given in 
the figure, the interchange rules of the Master Car Builders' 
Association authorize a road to refuse to accept a car from 
another road for transportation. At junction points of rail- 
roads inspectors are detailed to see that this rule (as well as 
many others) is complied with in respect to all cars offered 
for transfer. 

TRAIN-BRAKES. 

333. Introduction. Owing to the very general misappre- 
hension that exists regarding the nature and intensity of the 
action of brakes, a complete analysis of the problem is con- 
sidered justifiable. This misapprehension is illustrated by the 
common notion (and even practice) that the effectiveness of 
braking a car is proportional to the brake pressure, and there- 
fore a brakeman is frequently seen using a bar to obtain a 
greater leverage on the brake-wheel and using his utmost 
strength to obtain the maximum pull on the brake-chain while 
the car is skidding along with locked wheels. 

When a vehicle is moving on a track with a considerable 
velocity, the mass of the vehicle possesses kinetic energy of 
translation and the wheels possess kinetic energy of rotation. 
To stop the vehicle, this energy must be destro3^ed. The 
rotary kinetic energy will vary from about 4 to 8% of the 
kinetic energy of translation, according to the car loading 
(see § 347). On steam railroads brake action is obtained by 
pressing brake-shoes against car-wheel treads. As the brake- 
shoe pressure increases, the brake-shoes retard with increasing 
force the rotary action of the wheels. As long as the wheels 
do not slip or "skid'' on the rails, the adhesion of th& rails 
forces them to rotate with a circumferential velocity equal to 
the train velocity. The retarding action of the brake-shoe 



400 



RAILROAD CONSTRUCTION. 



§333. 




Fig. 204. — M. C. B. Standard Wheel-tread and Axle. 



§ 334. ROLLING-STOCK. 401 

checks first the rotative kinetic energy (which is small), and 
the remainder develops a tendency for the wheel to slip on the 
rail. Since the rotative kinetic energy is such a small per- 
centage of the total, it will hereafter be ignored, except as 
specifically stated, and it w411 be assumed for simplicity that 
the only w^ork of the brakes is to overcome the kinetic energy 
of translation. The possible effect of grade in assisting or 
preventing retardation, and the effect of all other track resist- 
ances, is also ignored. The amount of the developed force 
which retards the train movement is limited to the possible 
adhesion or static friction betw^een the wheel and the rail. 
When the friction between the brake-shoe and the wheel ex- 
ceeds the adhesion betw^een the w^heel and the rail, the wheel 
skids, and then the friction between the w^heel and the rail 
at once drops to a much less quantity. It must therefore be 
remembered at the outset that the retarding action of brake- 
shoes on wheels as a means of stopping a train is absolutely 
limited by the possible static friction between the braked 
wheels and the rails. 

334. Laws of friction as applied to this problem. Much of 
the misapprehension regarding this problem arises from a very 
conamon and widespread misstatement of the general law^s of 
friction. It is frequently stated that friction is independent 
of the velocity and of the unit of pressure. The first of these 
so-called laws is not even approximately true. A very exhaus- 
tive series of tests were made by Capt. Douglas Galton on the 
Brighton Railw^ay in England in 1878 and 1879, and by M. 
George Marie on the Paris and Lyons Railw^ay in 1879, w^ith 
trains which were specially fitted w ith train-brakes and wath 
dynagraphs of various kinds to measure the action of the 
brakes. Experience proved that variations in the condition of 
the rails (w^et or dry), and numerous irregularities incident to 
measuring the forces acting on a heavy body moving wdth a 
high velocity, w^ere such as to give somew^hat discordant re- 
sults, even when the conditions w^ere made as nearly identical 
as possible. But the tests were carried so far and so persist- 
ently that the general laws stated below were demonstrated 
beyond question, and even the numerical constants w^ere deter- 
mined as closely as they may be practically utilized. These 
law'S may be briefly stated as follows: 

(a) The coefficient of friction between cast-iron brake-blocks 



402 RAILROAD CDNSTRUCTION. § 334. 

and steel tires is about .3 when the wheels are "just mov- 
ing ^%* it drops to about .16 when the velocity is about 30 miles 
per hour, and is less than .10 when the velocity is 60 miles per 
hour. These figures fluctuate considerably with the condition 
of the rails, wet or dry. 

(b) The coefficient of friction is greatest w^hen the brakes 
are first applied; it then reduces very rapidly, decreasing 
nearly one third after the brakes have been applied 10 seconds, 
and dropping to nearly one half in the course of 20 seconds. 
Although the general truth of this law was estabhshed beyond 
question, the tests to demonstrate the law of the variation of 
friction with time of application were too few to determine 
accurately the numerical constants. 

(c) The friction of skidded wheels on rails is always very 
much less than the adhesion when the wheel is rolling on the 
rail — sometimes less than one third as much. 

(d) An analysis of the tests all pointed to a law that the 
friction developed does not increase as rapidly as the intensity 
of pressure increases, but this may hardly be considered as 
an estabhshed law. 

(e) The adhesion between the wheel and the rail appears to 
be independent of velocity. The adhesion here means the force 
that must be developed before the wheel will slip on the rail. 

The practical effect of these laws is shown by the following 
observed phenomena: 

(a) When the brakes are first applied (the velocity being 
very high), a brake pressure far in excess of the weight on the 
wheel (even three or four times as much) may be applied with- 
out skidding the wheel. This is partly due to the fact that 
the wheel has a very high rotative kinetic energy (w^hich varies 
as the square of the velocity, and which must be overcome 
first), but it is chiefly due to the fact that the coefficient of 
friction at the higher velocity is very small (at 60 miles per 
hour it is about .07), while the adhesion between the wheel and 
the rail is independent of the velocity. 

(6) As the velocity decreases the brake pressure must be 
decreased or the wheels will skid. Although the friction de- 
creases with the time required to stop and increases with the 
reduction of speed, and these two effects tend to neutralize 
each other, yet unless the stop is very slow, the increase in 
friction due to reduction of speed is much greater than the 



§ 835. ROLLING-STOCK. 403 

decrease due to time, and therefore the brake pressure must 
not be greater than the weight on the wheel, unless momentarily 
while the speed is still very high. 

(c) The adhesion between wheels and rails varies from .20 
to .25 and over when the rail is dry. When wet and slippery 
it may fall to .18 or even .15. The use of sand will always 
raise it aboA^e .20, and on a dry rail, when the sand is not blown 
away by wind, it may raise it to .35 or even .40. 

(d) Experiments were made With an automatic valve by 
which the brake-shoe pressure against the wheel should be 
reduced as the friction increased, but since (1) the essential 
requirement is that the friction produced by the brake-shoes 
shall not exceed the adhesion between rail and wheel, and 
since (2) the rail-wheel adhesion is a very variable quantity, 
depending on whether the rail is wet or dry, it has been found 
impracticable to use such a valve, and that the best plan is to 
leave it to the engineer to vary the pressure, if necessary, by the 
use of the brake-valve. 

MECHANISM OF BRAKES. 

335. Hand-brakes. The old style of brakes consists of brake- 
shoes of some type which al-e pressed against the wheel-treads 
by means of a brake-beam, which is operated by means of a 
hand-windlass and chain operating a set of levers. It is desir- 
able that brakes shall not be set so tightly that the wheels 
shall be locked, and then slide over the track, producing 
flat places on them, which are very destructive to the 
rolling-stock and track afterward, on accoimt of the impact 
occasioned at each revolution. With air-brakes the maximum 
pressure of the brake-shoes can be quite carefully regulated, 
and they are so designed that the maximum pressure exerted 
by any pair of brake-shoes on the wheels of any axle shall not 
exceed a certain per cent, of the weight carried by that axle 
when the car is empty, 90% being the figure usually adopted 
for passenger-cars and 70% for freight-cars. Consider the 
case of a freight-car of 100000 lbs. capacity, weighing 33100 lbs., 
or 8275 lbs. on an axle, and equipped with a hand-brake which 
operates the levers and brake-beams, which are sketched in 
Fig. 205. The dead weight on an axle is 8275 lbs.; 70% of 



404 



RAILROAD CONSTRUCTION. 



§335. 



this is 5792 lbs., which is the maximum allowable pressure 
per brake-beam, or 2896 lbs. per brake-shoe. With the dimen- 
sions shown, such a pressure will be produced by a pull of about 
1158 lbs. on the brake-chain. The power gained by the brake- 
wheel is not equal to the ratio of the brake-wheel diameter 
to the diameter of the shaft, about which the brake-chain 
winds, which is about 16 to 1§. The ratio of the circumfer- 
ence of the brake-wheel to the length of chain wound up by 
one complete turn would be a closer figure. The loss of effi- 




^--===^5792 



Fig. 205. — Sketch of Mechanism of Hand-brake. 



ciency in such a clumsy mechanism also reduces the effective 
ratio. Assuming the effective ratio as 6:1 it would require a 
pull of 193 lbs. at the circumference of the brake-wheel to 
exert 1158 lbs. pull on the brake-chain, or 5792 lbs. pressure 
on the wheels at J5, and even this will not lock the wheels when 
the car is empty, much less when it is loaded. Note that the 
pressures at A and B are unequal. This is somewhat objec- 
tionable, but it is unavoidable with this simple form of brake- 
beam. More complicated forms to avoid this are sometimes 
used. Hand-brakes are, of course, cheapest in first cost, and 
even with the best of automatic brakes, additional mechanism 
to operate the brakes by hand in' an emergency is always pro- 
vided, but their slow operation when a quick stop is desired 
makes it exceedingly dangerous to attempt to run a train at 
high speed unless some automatic brake directly under the 
control of the engineer is at hand. The great increase in the 



§ 337. ROLLING-STOCK* 405 

average velocity of trains during recent jesiTs has only been 
rendered possible by the invention of automatic brakes. 

336. "Straight" air-brakes. The essential constructive fea- 
tures of this form of brake are (1) an air-pump on the engine, 
operated by steam, which compresses air into a reservoir on 
the engine; (2) a ^'brake-pipe" running from the reservoir 
to the rear of the engine and pipes running under each car, 
the pipes having flexible connections at the ends of the cars 
and engine; (3) a cylinder and piston under each car which 
operates the brakes by a system of levers, the cylinder being 
connected to the brake-pipe. The reservoir on the engine 
holds compressed air at about 45 lbs. pressure. To operate the 
brakes, a valve on the engine is opened which allows the com- 
pressed air to flow from the reservoir through the brake-pipe 
to each cyhnder, mo^dng the piston, which thereby moves the 
levers and applies the brakes. The defects of this system are 
many: (1) With a long train, considerable time is required for 
the air to flow from the reservoir on the engine to the rear cars, 
and for an emergency-stop even this delay would often be 
fatal; (2) if the train breaks in two, the rear portion is not 
provided with power for operating the brakes, and a dangerous 
collision would often be the result; (3) if an air-pipe coupling 
bursts under any car, the ^whole system becomes absolutely 
helpless, and as such a thing might happen during some emer- 
gency, the accident would then be especially fatal. 

This form of brake has almost, if not entirely, passed out of 
use. It is here briefly described in order to show the logical 
development of the form which is now in almost universal use, 
the automatic. 

337. Automatic air-brakes. The above defects have been 
overcome by a method which may be briefly stated as follow^s: 
A reservoir for compressed air is placed under each car and the 
tender; whenever the pressure in these reservoirs is reduced 
for any reason, it is automatically replenished from the main 
reservoir on the engine; w^henever the pressure in the brake- 
pipe is reduced for any cause (opening a valve at any point of 
its length, parting of the train, or bursting of a pipe or coupler) , 
valves are automatically moved under each car to operate the 
piston and put on the brakes. All the brakes on the train are 
thus applied almost simultaneously. K the train breaks in two, 
both sections will at once have all the brakes apphed automati- 



406 



RAILROAD CONSTRUCTION. 



cally; if a coupling or pipe bursts, the brakes are at oncei 
and attention is thereby attracted to the defect; if a|byt 
gency should arise, such that the conductor desires I 
the train instantly without even taking time to signaf^^^ 
engineer, he can do so by opening a valve placed on el 
which admits air to the train-pipe, which will set theP 1 
on the whole train, and the engineer, being able to cltotkl 
instantly what had occurred, would shut off steam I 
whatever else was necessary to stop the train as quickly 1 
sible. The most important and essential detail of this i 
is the ''automatic triple valve" placed under each car. lUitol 
ing from the Westinghouse Air-brake Company's Instil f^l\ 
Book, '' A moderate reduction of air pressure in the trail m 
causes the greater pressure remaining stored in the aul is 
reservoir to force the piston of the triple valve and its p lbs 
valve to a position which will allow the air in the au^Lp 
reservoir to pass directly into the brake-cylinder and applpg 
brake. A sudden or violent reduction of the air in the tk 
pipe produces the same effect, and in addition causes sultri 
mental valves in the triple valve to be opened, permittingL 
pressure from the train-pipe to also enter the brake-cylirL 
augmenting the pressure derived from the auxiliary reseri 
about 20%, producing practically instantaneous action ofL 
brakes to their highest efficiency throughout the entire tr|i 
When the pressure in the brake-pipe is again restored tol 
amount in excess of that remaining in the auxiliary reserv 
the piston- and slide-valves are forced in the opposite direct 
to their normal position, opening communication from the trs 
pipe to the auxiliary reservoir, and permitting the air in 1 
brake-cyhnder to escape to the atmosphere, thus releasing t 
brakes. If the engineer wishes to apply the brake, he mo^ 
the handle of the engineer's brake-valve to the right, whi 
first closes a port, retaining the pressure in the main reserve 
and then permits a portion of the air in the train-pipe to escar 
To release the brakes, he moves the handle to the extrer 
left, which allows the air in the main reservoir to flow free 
into the brake-pipe, restoring the pressure therein." 

338. Tests to measure the efficiency of brakes. Let v repr( 
sent the velocity of a train in feet per second; W, its weighi 
Fy the retarding force due to the brakes; d, the distance in fee 
required to make a stop; and g, the acceleration of gravit 



). ROLLING-STOCK. 407 

) feet per square second) ; then the kinetic energy pos- 
. by the train (disregarding for the pTesent the rotative 

^jj,c energy of the wheels) =-,r — . The work done in stop- 

the train =i^c?. .'. Fd = ~^r—. The ratio of the retarding 
to the weight, 



I 

der to compare tests made under varying conditions, the 
|F-T- W should be corrected for the effect of grade ( + or — ), 
^^f, and also for the proportion of the weight of the train 
'V is on braked wheels. For example, a train w^eighed 
j6 lbs., the proportion on braked wheels was 67%, speed 
^^!3t per second, length of stop 450 feet, track level. Sub- 
jing these values in the above formula, we find (F~-W) 
J4. This value is really unduly favorable, since the ordi- 
I track resistance helps to stop the train. This has a value 
I'm 6 to 20 lbs. per ton, averaging say 10 lbs. per ton dur- 
ae stop, or .005 of the weight. Since the effect of this is 
and is nearly constant for all trains, it may be ignored 
mparative tests. The grade in -this case was level, and 
^fore grade had no effect. But since only 67% of the 
hi was on braked wheels, the ratio, on the basis of all the 
Us braked, or of the weight reduced to that actually on the 
d wheels, is 0.124^.67 = 0.185. This was called a ''good" 
although as high a ratio as 0.200 has been obtained. 
). Brake-shoes. Brake-shoes v»^ere formerly made of 
ght iron, but when it was discovered that cast-iron shoes 
1i answer the purpose, the use of v\TOught-iron shoes was 
ijioned, since the cast-iron shoes are so much cheeper. A 
^3 practice is to form the brake-shoe and its head in one 
, which is cheaper in first cost, but when the wearing-sur- 
ds too far gone for further use, the whole casting must be 
A^ed. The "Christie" shoe, adopted by the Master Car 
lers' Association as standard, has a separate shoe which 
;tened to the head by means of a wrought-iron key. The 
is bcA^eled J'' in a width of 3|'' to fit the coned wheel. 
is a greater bevel than the standard coning of a car-wheel, 
perhaps done to allow for some bending of the brake- 



408 RAILROAD CONSTRUCTION. § 339. 

beam and also so that the maximum pressure (and wear) should 
come on the outside of the tread, rather than next to the flange, 
where it might tend to produce sharp flanges. By concen- 
trating the brake-shoe wear on the outer side of the tread, the 
wear on the tread is more nearly equalized, since the rail wears 
the wheel-tread chiefly near the flange. This same idea is 
developed still further in the ^' flange-shoes," which have a 
curved form to fit the wheel-flange and which bear on the 
wheel on the flange and on the outside of the tread. It is 
claimed that by this means the standard form of the tread is 
better preserved than when the wear is entirely on the tread. 
The Congdon brake-shoe is one of a type in which wrought- 
iron pieces are inserted in the face of a cast-iron shoe. It is 
claimed that these increase the life of the shoe. 



CHAPTER XVI. 

TRAIN RESISTANCE. 

340. Classification of the various forms. The various resist- 
ances which must be overcome by the power of the locomotive 
may be classified as follows : 

(a) Resistances internal to the locomotive, which include fric- 
tion of the valve-gear, piston- and connecting-rods, journal 
friction of the drivers; also all the loss due to radiation, con- 
densation, friction of the steam in the passages, etc. In short, 
these resistances are the sum-total of the losses by which the 
power at the circumference of the drivers is less than the power 
developed by the boiler. 

(h) Velocity resistances, which include the atmospheric resist- 
ances on the ends and sides; oscillation and concussion resist- 
ances, due to uneven track, etc. 

(c) Wheel resistances, which include the rolling friction be- 
tween the wheels and the rails of all the wheels (including the 
drivers); also the journal friction of all the axles, except those 
of the drivers, 

(d) Grade and curve resistances, which include those resist- 
ances which are due to grade and to curves, and which are not 
found on a straight and level track. 

(e) Brake resistances. As shown later, brakes consume 
power and to the extent of their use increase the energy to 
be developed by the locomotive. 

(/) Inertia resistances. The resistance due to inertia is not 
generally considered as a train resistance because the energy 
which is stored up in the train as kinetic energy ma}^ be util- 
ized in overcoming future resistances. But in a discussion 
of the demands on the tractive power of the engine, one of the 
chief items is the energy required to rapidly give to a starting 
train its normal velocity. This is especially true of suburban 
trains, which must acquire speed very quickly in order that 

409 



410 RAILROAD CONSTRUCTION. ^ § 341. 

their general average speed between termini may be even reason- 
ably fast. 

341. Resistances internal to the locomotive. These are re- 
sistances which do not tax the adhesion of the drivers to the 
rails, and hence are frequently considered as not being a part 
of the train resistance properly so called. If the engine were 
considered as lifted from the rails and made to drive a belt 
placed around the drivers, then all the power that reached the 
belt would be the power that is ordinarily available for adhe- 
sion, while the remainder would be that consumed internally 
by the engine. The power developed by an engine may be 
obtained by taking indicator diagrams which show the actual 
steam pressure in a cylinder at any part of a stroke. From 
such a diagram the average steam pressure is easily obtained, 
and this average pressure, multiplied by the length of the stroke 
and by the net area of the piston, gives the energy developed 
by one half-stroke of one piston. Four times this product 
divided by 550 times the time in seconds required for one stroke 
gives the '^indicated horse-power" Even this calculation 
gives merely the power behind the piston, which is several per 
cent, greater than the power which reaches the circumference 
of the drivers, owing to the friction of the piston, piston-rod, 
cross-head, connecting-rod bearings, and driving-wheel jour- 
nals. (See § 322, Chapter XV.) By measuring the amount 
of water used and turned into steam, and by noting the boiler 
pressure, the energy possessed by the steam used is readily 
computed. The indicator diagrams will show the amount of 
steam that has been effective in producing power at the cylin- 
ders. The steam accounted for by the diagrams will ordinarily 
amount to 80 or 85% of the steam developed by the boiler, 
and the other 15 or 20% represents the loss of energy due to 
radiation, condensation, etc. From actual tests it has been 
found that the power consumed by an engine running light is 
about 11%, of that required by the engine when working hard 
in express freight service. But since the engine resistances 
(friction, etc.) are increased when it is pulling a load, it was 
estimated, after allowing for this fact, that about 15 or 16% 
of the power developed by the pistons was consumed by the 
engine, leaving about 84 to 85% for the train. i 

342. Velocity resistances, (a) Atmospheric, This consists of 
the head and tail resistances and the side resistance. The head 



§ 343. TRAIN RESISTANCE. 411 

and tail resistances are neaily constant for all trains qf given 
velocity, varying but slightly with the varying cross-sections 
of engines and cars. The side resistance varies with the length 
of the train and the character of the cars, box-cars or flats, etc. 
Vestibuling cars has a considerable effect in reducing this side 
resistance by preventing much of the eddying of air-currents 
between the cars, although this is one of the least of the ad 
vantages of vestibuling. Atmospheric resistance is generally 
assumed to vary as the square of the velocity, and although 
this may be nearly true, it has been experimentally demon 
strated to be at least inaccurate. The head resistance is gen 
erally assumed to vary as the area of the cross-section, but this 
has been definitely demonstrated to be very far from true. A 
freight-train composed parth^ of flat-cars and partly of box- 
cars will encounter considerably more atmospheric resistance 
than one made exclusively of either kind, other things being 
equal. The definite information on this subject is very luisat- 
isfactory, but this is possibly due to the fact that it is of little 
practical importance to know just how much such resistance 
amounts to. 

(b) Oscillatory and concussive. These resistances are con- 
sidered to vary as the square of the velocity. Probably this 
is nearly, if not quite, correct on the general principle that such 
resistances are a succession of impacts and the force of impacts 
varies as the square of the velocity. These impacts are due to 
the defects of the track, and even though it were possible to 
make a precise determination of the amount of this resistance 
in any particular case, the value obtained would only be true 
for that particular piece of track and for the particular degree 
of excellence or defect which the track then possessed. The 
general improvement of track maintenance during late years 
has had a large influence in increasing the possible train-load 
by decreasing the train resistance. The expenditure of money 
to improve track will give a road a large advantage over a 
competing road with a poorer track, by reducing train resist- 
ance, and thus reducing the cost of handling traffic. 

343. Wheel resistances, (a) Rolling friction of the wheels. 
To determine experimentally the rolling friction of wheels, 
apar-t from all journal friction, is a very difficult matter and 
has never been satisfactorily accomplished. Theory as well 
as practice shows that the higher and the more perfect the 



412 RAILROAD CONSTRUCTION. §343^ 

elasticity of the wheel and the surface, the less will be the roll- 
ing friction. But the determination, if made, would be of 
theoretical interest only. 

The combined effect of rolling friction and journal friction 
is determinable with comparative ease. From the nature of 
the case no great reduction of the rolling friction by any device 
is possible. It is only a very insignificant part of the total 
train resistance. 

(h) Journal friction of the axles. This form of resistance has 
been studied quite extensively by means of the measurement 
of the force required to turn an axle in its bearings under 
various conditions of pressure, speed, extent of lubrication, 
and temperature. The following laws have been fairly well 
established: (1) The coefficient of friction increases as the pres- 
sure diminishes; (2) it is higher at very slow speeds, gradually 
diminishing to a minimum at a speed corresponding to a train 
velocity of about 10 miles per hour, then slowly increasing 
with the speed; it is very dependent on the perfection of the 
lubrication, it being reduced to one sixth or one tenth, when the 
axle is lubricated by a bath of oil rather than by a mere pad 
or wad of waste on one side of the journal; (3) it is much lower 
at higher temperature, and vice versa. The practical effect of 
these laws is shown by the observed facts that (1) loaded cars 
have a less resistance per ton than unloaded cars, the figures 
being (for speeds of about 10 to 20 miles per hour) : 

For passenger- and loaded freight-cars. . . 4 lbs. per ton 

^ ' empty freight-cars 6 ' ^ '' " 

'' street-cars 10 '' '' '' 

'' freight-trucks without load 14 '' '' '' 

(2) When starting a train, the resistances are about 20 lbs. 
per ton, notwithstanding the fact that the velocity resistances 
are practically zero; at about 2 miles per hour it will drop to 
10 lbs. per ton and above 10 miles per hour it may drop to 
4 lbs. per ton if the cars are in good condition. (3) The re- 
sistance could probably be materially lowered if some practicable 
form of journal-box could be devised which would give a more 
perfect lubrication. (4) It is observed that freight-train loads 
must be cut down in winter by about 10 or 15% of the loads 
that the same engine can haul over the same track in summer. 
This is due partly to the extra roughness and inelasticity of the 



§344. 



TRAIN RESISTANCE. 



413 



track in winter, and partly to increased radiation from the 
engine wasting some energy, but this will not account for all 
of the loss, and the effect, which is probably due largely to the 
lower temperature of the journal-boxes, is very marked and 
costly. It has been suggested that a jacketing of the journal- 
boxes, which would prevent rapid radiation of heat and enable 
them to retain some of the heat developed by friction, would 
result in a saving amply repaying the cost of the device. 

Roller journals for cars have been frequently suggested, and 
experiments have been made with them. It is found that they 
are very effective at low velocities, greatly reducing the start- 
ing resistance, which is very high with the ordinary forms of 
journals. But the advantages disappear as the velocity in- 
creases. The advantages also decrease as the load is increased, 
so that with heavily loaded cars the gain is small. The excess 
of cost for construction and maintenance has been found to be 
more than the gain from power saved. 

344. Grade resistance. The amount of this may be com- 
puted with mathematical exactness. Assume that the bail 
or cylinder (see Fig. 206) is being drawn up the plane. If W 




Fig. 206. 

is the weight, N the normal pressure against the rail, and G 

the force required to hold it or to draw it up the plane with 

uniform velocity, the rolling resistances being considered zero 

or considered as provided for by other forces, then 

Wh 
G:W::h:d, or G=-^) 

but for all ordinary railroad grades, c? = c to wdthin a tenth of 

Wh 
1%, i.e., G = = TT X rate of grade. In order that the student 

may appreciate the exact amount of this approximation the per- 
centage of slope distance to its horizontal projection is given in 
the following tabular form: 



414 



RAILROAD CONSTRUCTION. 



§344. 



Grade in per cent. 


1 


2 


3 


1 


5 


Slope dist.^jQp 

nor. dist. 


100.005 


100.020 


100.045 


100.080 


100.125 



Grade in per cent. 


6 


7 


8 


9 


10 


Slonedist.^^Qp 

nor. disi. 


100.180 


100.245 


100.319 


100.404 


100.499 



This shows also the error on various grades of measuring with 
the tape on the ground rather than held horizontally. Since 
almost all railroad grades are less than 2% (where the error 
is but .02 of 1%), and anything in excess of 4% is unheard 
of for normal construction, the error in the approximation 
is generally too small for practical consideration. 

If the rate of grade is 1 : 100, G = WXjio, i-e., G=^2Q lbs. 
per ton ; . '. for any per cent, of grade, G ^ (20 X per cent, of grade) 
pounds per ton. When moving up a grade this force G is to 
be overcome in addition to all the other resistances. When 
moving down a grade, the force G assists the motion and may 
be more than sufficient to move the train at its highest allow- 
able velocit3\ The force required to move a train on a level 
track at ordinary freight-train speeds (say 20 miles per hour) 
is about 7 lbs. per ton. "^ A down grade of ^^ of 1% will fur- 
nish the same power; therefore on a down grade of 0.35%, a 
freight-train would move indefinitely at about 20 miles per hour. 
If the grade were higher and the train were allowed to gain 
speed freely, the speed w^ould increase until the resistance at 
that speed would equal W times the rate of grade, when the 
velocity would become uniform and remain so as long as the 
conditions w^ere constant. If this speed was higher than a 
safe permissible speed, brakes must be applied and power 
wasted. The fact that one terminal of a road is considerably 
higher than the other does not necessarily imply that the extra 
power needed to overcome the difference of elevation is a 
total waste of energy, especially if the maximum girades are 
so low that brakes will never need to be applied to reduce a 
dangerously high velocity, for although more power must be 



§ 347. TRAIN RESISTANCE. 415 

used in ascending the grades, there is a considerable saving of 
power in descending the grades. The amount of this saAdng 
will be discussed more fully in Chapter XXIII. 

345. Curve resistance. Some of the principal laws will be 
here given without elaboration. A more detailed discussion 
wall be given in Chapter XXII. 

(a) While the total curve resistance increases as the degree 
of curve increases, the resistance j)er degree of curve is much 
greater for easy curves than for sharp curves; e.g.j the resist- 
ance on the excessively sharp curves (radius 90 feet) of the 
elevated roads of New York City is very much less per degree 
of curve than that on curves of 1° to 5°. (h) Curve resistance 
increases wdth the velocity, (c) The total resistance on a 
curve depends on the central angle rather than on the radius; 
I.e., two curves of the same central angle but of different radius 
would cause about the same total curve resistance. This is 
partly explained by the fact that the longitudinal slipping will 
be the same in each case. (See § 311, Chapter XV.) In each 
case also the trucks must be twisted around and the w^heels 
slipped laterally on the rails by the same amount J"^. (See 
§ 312, Chapter XV.) 

346. Brake resistances. If a down grade is excessively steep 
so that brakes must be applied to prevent the train acquiring 
a dangerous velocity, the energ\^ consumed is hopelessly lost 
without any compensation. When trains are required to make 
frequent stops and yet maintain a high average speed, consid- 
erable power is consumed by the application of brakes in stop- 
ping. All the energy which is thus turned into heat is hope- 
lessly lost, and in addition a very considerable amount of steam 
is drawn from the boiler to operate the air-brakes, which con- 
sume the power already developed. It can be easily demonstrated 
that engines drawing trains in suburban service, making fre- 
quent stops, and yet developing high speed between stops, will 
consume a very large proportion of the total powxr developed 
by the use of brakes. Note the double loss. The brakes con- 
sume power already developed and stored in the train as kinetic 
or potential energy, while the operation of the brakes requires 
additional steam power from the engine. 

347. Inertia resistance. The two forms of train resistance 
which under some circumstances are the greatest resistances 
to be overcome by the engine are the grade and inertia resist- 



416 EAILROAD CONS'TRtrcTlON. § 347. 

ances, and fortunately both of these resistances may be com- 
puted with mathematical precision. The problem may be 
stated as follows: What constant force P (in addition to the 
forces required to overcome the various frictional resistances, 
etc.) will be required to impart to a body a velocity of v feet 
per second in a distance of s feet? The required number of 
foot-pounds of energy is evidently Ps, But this work imparts 

a kinetic energy which may be expressed by -^— . Equating 
these values, we have Ps=-7r— , or 

^=^ <138) 

The force required to increase the velocity from Vj to v^ may 

W 
likewise be stated as P=7^ — (^2^— '^i^)- Substituting in the 

Zgs 

formula the values TF=2000 lbs. (one ton), gr =32.16, and s = 

5280 feet (one mile), we have 

P = .00588(V-'^'iO. 
Multiplying by (5280 -^ 3600)2 ^^ change the unit of velocity 
to miles per hour, we have 

P = .01267(F22-yi2). 

But this formula must be modified on account of the rotative 
kinetic energy which must be imparted to the wheels of the cars. 
The precise additional percentage depends on the particular 
design of the cars and their loading and also on the design of 
the locomotive. Consider as an example a box-car, 60000 lbs. 
capacity, weighing 33000 lbs. The wheels have a diameter 
of 36'' and their radius of gyration is about 13''*. Each wheel 
weighs 700 lbs. The rotative kinetic energy of each wheel is 
4877 ft.-lbs. when the velocity is 20 miles per hour, and for 
the eight wheels it is 39016 ft.-lbs. For greater precision 
(really needless) we may add 192 ft.-lbs. as the rotative kinetic 
energy of the axles. When the car is fully loaded (weight 
93000 lbs.) the kinetic energy of translation is 1,244,340 ft.-lbs.; 
when empty (weight 33000 lbs.) the energy is 441540 ft.-lbs- 
The rotative kinetic energy thus adds (for this particular 
car) 3.15% (when the car is loaded) and 8.9% (when the car 
is empty) to the kinetic energy of translation. The kinetic 



§ 347. TRAIN RESISTANCE. 417 

energy which is similarly added, owing to the rotation of the 
Vv^heels and axles of the locomotive, might be simiraiiy com- 
puted. For one type of locomotive it has been figured at about 
8%. The variations in design, and particularly the fluctua- 
tions of loading, render useless any great precision in these 
computations. For a train of '' empties'' the figure would be 
high, probably 8 to 9%; for a fully loaded train it will not 
much exceed 3%. Wellington considered that 6% is a good 
average value to use (actually used 6.14% for '^ease of compu- 
tation"), but considering (a) the increasing proportion of live 
load to dead load in modern car design, (6) the greater care 
now used to make up full train-loads, and (c) the fact that 
fidl train-loads are the critical loads, it would appear that 5% 
is a better average for the conditions of modern practice. Even 
this figure allows something for the higher percentage for the 
locomotive and something for a few empties in the train. There- 
fore, adding 5% to the coefficient in the above equation, we 
have the true equation 

P = .0133(PV-"i^i'), (139') 

in which Fj ^'^^ ^i ^^^ ^^^ higher and lower velocities respec- 
tively in miles per hour, and P is the force required per ton to 
impart that difference of velocity in a distance of one mile 
If more convenient, the formula may be used thus: 

P:=^^(TV-Fi^), .... (140) 

in which s is the distance in feet and Pi is the corresponding 
force. 

As a numerical illustration, the force required per ton to 
impart a kinetic energy due to a velocity of 20 miles per hour 
in a distance of 1000 feet will equal 

70.224(400-0) _ 
Pi— ^00 -28 lbs., 

which is the equivalent (see § 344) of a 1.4% grade. Since the 
velocity enters the formula as V^, while the distance enters only 
in the first power, it follows that it will require four times the 
force to produce twice the velocity in the same distance, or 
that with the same force it will require four times the distance 
to attain twice the velocity. 



418 RAILROAD CONSTRUCTION. § 347c 

As another numerical illustration, if a train is to increase its 
speed froni 15 miles per hour to 60 miles per hour in a distance 
of 2000 feet, the force required (in addition to all the other 
resistances) will be 

^ 70.224(3600-225) , , o rmu 4. 

' ^ ^000 = 1 18.50 lbs. per ton. 

This is equivalent to a 5.9% grade and shows at once that it 
would be impossible unless there were a very heavy doT\Ti 
grade, or that the train was very light and the engine very 
powerful. 

348. Dynamometer tests. These are made by putting a 
''dynamometer-car'' between the engine and the cars to be 
tested. Suitable mechanism makes an automatic record of 
the force which is transmitted through the dynamometer at 
any instant, and also a record of the velocity at any instant. 
One of the practical difficulties is the accurate determination 
of the velocity at any instant when the velocity is fluctuating. 
When the velocity is decreasing, the kinetic energy of the train 
is being turned into work and the force transmitted through the 
dynamometer is less than the amount of the resistance which 
is actually being overcome. On the other hand, when the 
velocity is increasing, the dynamometer indicates a larger 
force than that required to overcome the resistances, but the 
excess force is being stored up in the train as kinetic energy. 
Grade has a similar effect, and the force indicated by the dy^- 
namometer may be greater or less than that required at the 
given velocity on a level by the force which is derived from, 
or is turned into, potential energy. Therefore the resistance 
indicated bj^ the dynamometer of a train will not be that on a 
level track at uniform velocity, unless the track is actually - 
level and the velocity really uniform. ■{ 

Dynamometer tests under other circumstances are there- 
fore of no value unless it is possible to determine the true 
velocity at any instant and its rate of change, and also to de- 
termine the grade. Of course, the grade is easily found. An 



350. 



TRAIN RESISTANCE. 



410 



illowance for an increase or decrease of kinetic or potential 
energy must therefore be made before it is possible to know 
[low much force is being spent on the ordinary resistances. 

349. Gravity or " drop " tests. Dynamometer tests require 
:he use of a dynamometer which is capable of measuring a 
■orce of several thousands of pounds, and which therefore 
?annot determine such values with a close percentage of accu- 
racy, especially if the force is small. A drop test utihzes the 
•orce of gra\dty which may be measured with mathematical 
accuracy. The general method is to select a stretch of track 
Nhich has a imiform grade of about 0.7% and which is prefer- 
ibly straight for tw^o or three miles. On such a grade cars 
^'ith running gear in good condition may be started by a push, 
rhe velocity will gradually increase until at some velocity, 
iepending on the resistances encountered, the cars will move 
iniformly. The only work requiring extreme care with this 
nethod is the determination of the velocity. If the velocity 
s fluctuating, as it is during the time when it is of the greatest 
mportance to know the velocity, it is not sufficient to deter- 
nine the time required to run some long measured distance, 
'or the average velocity thus obtained would probably differ 




Fig. 207. — Loss in Velocity-head. 



jonsiderably from the velocity at the beginning and end of that 
5pace. If the train consists of five cars or more, the velocity 
nay be determined electrically (as described by Welhngton 
n his *' Economic Location," etc., p. 793 et seq.) from the 
lutomatic record made on a chronograph of the passage of the 
irst wheel and thje last, the chronograph also recording auto- 



420 RAILROAD CONSTRUCTION § 350. 

matically the ticks of a clock beating seconds. From this the 
exact time of the passage of the first and last wheels of the 
train of cars may be determined to the tenth or twentieth of a 
second. 

Velocity -head. From theoretical mechanics we know that 
if a body descends through any path by the action of gravity, 
and is unaffected by friction, its velocity at any point in the 
direction of the path of motion is V =\/2gli. If the body is 
retarded by resistances, its velocity at any point will be less 
than this. If AM, Fig. 207, represents any grade (exaggerated 
of course), then BJ, CK, etc., represent the actual fall at any 

point. Let BF represent the fall /ij, determined from hi = ~, 

in which v^ is the actual observed velocity at J. Then Ji^= the 
velocity-head consumed by the resistances between A and J. 
If the train continues to K, the corresponding /lo is CG; .the 
remaining fall GK consists of GN^ (=JF, which is the velocity- 
head lost back of J) and NKj the velocity-head lost between J 
and K. At some velocity (Vn) on any grade, the velocity 
will not further increase and the line AFGHI will then be hori- 
zontal and at a distance (Jin) = EI below A , , ,E, The grade 
AM is the "grade of repose'' for that velocity (Fn); i.e., it is 
the grade that would just permit the train to move indefinitely 
at the velocity Vn- The broken line AFGHI should really be 
a curve, and the grade of repose at any point is the angle between 
AM and the tangent to that curve at the given point. The 
"grade of repose" by its definition gives the total resistance 
of the train at the particular velocit}^, or multiplying the grade 
of repose in per cent, by 20 gives the pounds per ton of resist- 
ance. Thus being able to determine the total resistance in 
pounds per ton at any velocity, the varia,tion of total resistance 
with velocity may be determined, and then by varying the 
resistances, using different kinds of cars, empty and loaded, 
box-cars and flats, the resistances of the different kinds at 
various velocities may be determined. 

350. Formulae for train resistance. These are generally given 
in one of the forms 



ji 



R = aV^-c, ... (1) 
R = hV^ + c, ... (2) 
R^aV^-hV^-^-c, . • (3) 



. . (141) 



§ 350. TKAIN RESISTANCE. 421 

in which R is the resistance in pounds per ton, a and h are coeffi- 
cients to be determined, Y is the velocity in miles per hour, and 
c is a constant, also to be determined. These formulae disregard 
grade and curve resistances, inertia resistance and the active 
resistance (or assistance) of windy as distinct from mere atmos- 
pheric resistance. In short, they are supposed to give the re- 
sistance of a train moving at a uniform velocity over a straight 
and level track, there being no appreciable wind. 

The various formulae are sometimes based directly on experi- 
ments made by the proposer of the formula ; sometimes they are 
deduced from a mere study of the results of one or more series 
of tests made by others. Unfortunately for either method, no 
one investigator has ever been able to make tests which are so 
thorough and made under such a wide range of conditions that 
his results may be considered as conclusive, while a student of 
the tests of others is handicapped by a lack of knowledge of 
precise conditions, which, if fully understood, would perhaps 
permit some reconciliation of the very discordant figures which 
are reported. As already intimated, the condition of the 
rolling stock, the unit weight on the axles, the lubrication of the 
axles, the length of the train in relation to its weight and the 
condition of the track, which uiyolves the weight of rail, spacing 
and size of ties, tamping of ties, etc., all have their influence in 
modifying the apparent resistance. There is also good reason to 
believe that the effect of grade, curvature, and changing velocity 
has not been properly allowed for in deducing many of the 
formulae. In view of all these considerations, it may be con- 
sidered as demonstrated that no one formula, and especially a 
simple formula, will represent the resistance for all conditions. 
But, since some of the calculations of railroad economics are 
absolutely dependent on the law of tractive resistance, some 
law must be deduced with sufficient accuracy for the purpose. 
Fortunately several of the formulae are amply accurate for such 
purposes. A report of a committee of the A. R. E. & M. W. 
Assoc. (1907) quoted sixty-one different formulae which have been 
suggested. Some of these are chiefly of historical value, since 
they were deduced from tests made many years ago with track 
and rolling stock very dissimilar from those in use at the present 
time. Such formulae will therefore be omitted. For con- 
venience of comparison, all formulae will be changed (if neces- 
sary) from the original statement of them so that they give the 



422 RAILROAD CONSTRUCTION. § 350. 

resistance per ton of 2000 pounds. The coefficients of V and 
V^ will be given decimally- Other notation occasionally used 
is as follows: 

^ = weight of train in tons of 2000 pounds; 
L = length of train in feet ; 
n = number of cars in train; 
A = area of front of train in square feet. 

(a) FormulsB of the first class: R = aV + c, Among those 
most commonly used are the following: 

Engineering News, E = 0.257 + 2.0 (142) 

Baldwin locomotive, E = 0.177 + 3.0 (143) 

New York Central, 7^ = 0.117+1.8, (144) 

Henderson, 2^ = 0.257 + ^- +0.5 (145) 

Although Henderson's formula is in a class by itself, on account 
of the extra term, and although it is not applicable to general 
use, when the character of the trains cannot be estimated, it 
is perhaps more accurate than the others. It is apparently not 
intended for use at very low velocities. 

(b) Formulae of the second class: R = bV^ + c: 

Crawford, i2 = 0.0021472 + 2.5 (146) 

Wolff, 7^ = 0.0035772 + 2.7 (147) 

Henderson, 7^ = 0.0046172 + 3.0 (148) 

Forney, 7^ = 0.0058572 + 4.0 (149) 

7^ = 0.005672 + ^-— +3.9 (for loaded flat cars) 



Wel- 
ling- { 
ton 



6472 
R = 0.007572 + — — + 3.9 (for loaded box cars) 

5772 
72 = 0.008372 + ^—7- +6.0 (for empty flat cars) 

6472 
72 = 0.010672 + ^-— +6.0 (for empty box cars) 



y. (150) 



Notice in formulae (150) the additioMal journal resistance 
(indicated by the constant term) for unloaded cars. The second 



§ 350. TRAIN RESISTANCE. 423 

term evidently indicates the atmospheric resistance. The first 
term allows for the oscillatory resistances. Assuming the con- 
stant term and the coefficients to have been correctly deter- 
mined, these formulae should be better than the others, since 
a choice of formulae can be made depending on the conditions. 
A train consisting partly of box-cars and partly of flat-cars 
will have a higher resistance than is shown by any of the above 
formulae (and not a mean value), on account of the increased 
atmospheric resistance acting on the irregular form of the train, 
(c) Formulae of the third class: R = aV-hbV^ + c: 

W.N.Smith, J^=ai77+ ^-^^^p +3.0; . . . . (151) 

VonBorries, i2 = 0.047 + 0.001672 H- 3.0; .... (152) 

4 87^ 
Lundie, i^ = 0.247+^^+4.0; (153) 

Sprague, 2^=0.177+ ''^^ +4.0 (154) 

Although several formulae have been proposed which involve 
the area of the front of the train in order to allow more definitely 
for the atmospheric resistance, only one of these (151) has been 
quoted. In applying this formula, the proper value to choose 
for A is somewhat indefinite, since the shape of the front of the 
train will make a considerable difference in the atmospheric 
resistance encountered. The area will vary from 80 to 100 
square feet. In the comparison of the formulae given below, 
A will be assumed as 100 square feet. In order to compare 
these resistances, the values of R for the various speeds of 10, 
20, 30, 40, 50, and 60 miles per hour will be computed by 
these formulae on the basis of a train of twelve cars, having a 
length of 480 feet, and a weight of 600 tons. Therefore in 
applying the formula, t = 600, L = 480, n = 12, and A = 100. In 
order to apply formula (150) to this case, it will be assumed 
that this train consists of loaded box-cars, and therefore we 
must apply the second of that group of formulae. Computing 
the resistance according to these several formulae, we may 
tabulate the results as given below; 



424 



KAILROAD CONSTRUCTION. 



§350. 



Formula. 


Velocity in miles per hour. 




10 


20 


30 


40 


50 


60 


142 
143 
144 
145 

146 

V 147 

\ 148 

149 

150 

151 
152 
153 
154 


2700 
2800 
1747 
2400 

1628 
1834 

2077 
2751 
2854 

2845 
2136 
4320 
3453 


4200 
3800 
2413 
3900 

2014 
2477 
2906 
3804 
4396 

3940 
2664 
7200 
4573 


5700 
4800 
3080 . 
5400 

2656 
3548 
4289 
5559 
6966 

5085 

3384 

11040 

5760 


7200 
5800 
3747 
6900 

3554 
5047 
6226 
8116 
10564 

6280 

4296 

15840 

7013 


8700 
6800 
4413 
8400 

4710 

6975 

8715 

11175 

15188 

7525 

5400 

19440 

8333 


10200 
7800 
5080 
9900 

6122 

9331 

11746 

15036 

20844 

8820 

6696 

28080 

9720 



Although there is a fair agreement among the results for 
ordinary velocities^ it should be said, in fairness to the proposers 
of the various formulae, that some of them evidently were not 
designed for use at high velocities such as 60 miles per hour. 

Another method of comparing formulae is to plot them on 
cross-section paper, using velocities as abscissae and resistances 
as ordinates. For general use this method may only be applied 
to formulae which do not involve the weight, length or area of 
the train nor the number of ears. All of the above formulae 
have thus been plotted on Plate IX, with the exception of Nos. 
145, 150, 151, 153, and 154. 



§ 350. 



TRAIN RESISTANCE. 



425 



— ^25- 
































PLA 


TE 


X. 


TRA 


IN F 


ESli 


3TAI 


MCE 




/ 




























/ 




























/ 


























1 


1 




—20 






















/ 


























/ 


1 


























c* 


/ 




























A 
























/a 


/ 


/ 




•2-15 


















/ 


'^"^ 


// 


i^ 




















/ 


J 


^ 


A 




















/ 


/ ^ 


/ 


/ 


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a 
















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A"/ 


i 


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'i 
















/J 


t/^' 


/ 


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1-10 

1 














/ 






/ 




/ 














A 


/ 


A 






/^ 


7 

/5^ 
















// 


^ 




# 


^ 


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/; 


^ 


/ 


/ 


X^ 






<^^ 












/^ 


^j 




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V 


/^ 


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y 


^ 


















/^ 


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^"^ 


















































































1 





2 


locit: 


30 
' in miles i 

.J ,-. 


er hour 


" 


60 



CHAPTER XVII 

COST OF RAILROADS. 

351. General considerations. Although there are many ele- 
ments in the cost of raihoads which are roughly constant per 
mile of road, yet the published reports of the cost of railroads 
differ very widely. The variation in the figures is due to several 
causes, (a) Economy requires that a road shall be operated 
and placed on an earning basis as soon as possible. Therefore 
the reported cost of a road during the first few years of its 
existence is somewhat less than that reported later. This is 
well illustrated when a long series of consecutive reports from 
an old-established road is available; nearly every year there 
will be shown an addition to the previous figures. And this 
is as it should be. The magnificent road-beds of some old 
roads cannot be the creation of a single season. It takes many 
years to produce such settled perfect structures. (6) A large 
part of the variation is due to a neglect to charge up " permanent 
improvements'' as additions to the cost of the road. For the 
first few years of the life of a road a great deal of work is done 
which is in reality a completion of the w^ork of construction, 
and yet the cost of it is buried under the item "maintenance 
of way.'' For example, a long wooden trestle is replaced by 
an earth embankment and a culvert. Since the original trestle 
is to be considered a temporary structure, the excess of the 
cost of the permanent structure over that of the temporary 
structure should evidently be considered as an addition to the 
cost of the road. But if the fiUing-in was done slowly, a few^ 
train-loads at a time, and the work scattered over man^^ years, 
the cost of operating the "mud-train" has perhaps been buried 
under "maintenance" charges, (c) The reports from w^hich 
many of the following figures were taken have not always 
analyzed the items of cost with the same detail as has been 
here attempted, and to that is probably due many of the A^aria- 
tions and apparent discrepancies 

426 



§ 352 COST OF EAILROADS. 427 

The various items of cost will be classified as follows: 

1. Preliminary financiering. 

2. Surveys and engineering expenses. 

3. Land and land damages. 

4. Clearing and grubbing. 

5. Earthwork. 

6. Bridges, trestles, and culverts 

7. Trackwork. 

8. Buildings and miscellaneous structures. 

9. Interest on construction. 
10. Telegraph line. 

352. Item I. PRELIMINARY FINANCIERING. The COst of this 
preliminary work is exceedingly variable. The work includes 
the clerical and legal work of organization, printing, engraving 
of stocks and bonds, and (sometimes the most expensive of all) 
the securing of a charter. This sometimes requires special 
legislative enactments, or may sometimes be secured from a 
State railroad commission. It has been estimated that about 
2% of the railway capital of Great Britain has been spent in 
Parliamentary expenses over the charters. These expenses 
are usually but a small percentage of the total cost of the enter- 
prise, but for important lines the gross cost is large, while the 
amount of money thus spent by organizations which have 
never succeeded in constructing their roads is, in the aggregate, 
an enormous amount, although it is of course not ascertainable 
by any investigator. 

Another occasional feature of the financing of a road must be 
kept in mind. The promoters of a railroad enterprise frequently 
endeavor to limit their own personal expenditures to the purely 
preliminary expenses as mentioned above. The project, after 
having been surveyed, mapped, and written up in a glowing 
"prospectus," is submitted to capitalists, in the endeavor to 
have them furnish money for construction, the money to be 
secured by bonds. If the project will stand it, the amount of 
the bond issue is made sufficient to pay the entire cost of the 
road, even with a discount of perhaps 15%. The bond issue 
may also provide for a very generous commission to the broker 
who is the intermediary between the promoters and the capi- 
talists. The bond issue may even provide for repaying the 
promoters for their preliminary expenses. Frequently a con- 
siderable proportion of the capital stock goes to the capitalists 



428 BAILHOAD OONSTHUCTION. § 352. 

who take the bonds, the promoters retaining only such propor- 
tion as may be agreed upon. In such a case, the capital stock 
is "pure velvet,'' and costs nothing. Its future value, whatever 
it may be, is so much clear profit. The effect of such a financial 
policy is to burden the project with a capitalization which is 
far in excess of the actual cost of constructing the road. Com- 
paratively few projects will stand such over-capitalization. 
The apparent financial failure of many railroads, which have 
gone into the hands of receivers is due to their inability to 
make returns on an over-capitalization rather than because 
they could not earn enough to pay the legitimate cost of their 
construction. These features of financiering are really foreign 
to the engineer's work, but he should know that many projects 
which would return a handsome profit on an investment amount- 
ing only to the legitimate cost, will be rejected by capitalists 
because it is apparent that there is not enough *^ velvet'' 
in it. 

353. Item 2. Surveys and Engineering Expenses. The 
comparison of a large number of itemized reports on the cost 
of construction shows that the cost of the "engineering^' will 
average about 2% of the total cost of construction. This in- 
cludes the cost of surveys and the cost of laying out and super- 
intending the constructive work. The cost of mere surveying 
up to the time when construction actually commences has 
been variously quoted at $60, $75, and even $150 per mile. 
In exceptional cases the surveying for a few miles through some 
gorge might cost many times this amount, but $150 per mile 
may be considered an ordinary maximum for difficult country. 
On the other hand, much construction has been done over the 
western prairies after hasty surveys costing not much over 

,$10 per mile. In the estimate given at the end of this chapter 
the cost of "engineering and office expenses" is given at 5% of 
the cost of the construction work. The item then includes the 
cost of the very considerable amount of clerical work and 
superintendence incident to the expenditure of such a large 
sum of money. 

354. Item 3. Land and Land Damages. The cost of this 
item varies from the extreme, in which not only the land for 



§ 355. COST OF RAILROADS. 429 

right-of-way but also grants of public land adjoining the road 
are given to the corporation as a subsidy, to the other extreme, 
where the right-of-way can only be obtained at exorbitant 
prices. The width required is variable, depending on the 
width that may be needed for deep cuts or high fills, or the 
extra land required for yards, stations, etc. A strip of. land 
1 mile long and 8.25 feet wide contains precisely 1 acre. An 
average width of 4 rods (66 feet), therefore, requires 8 acres per 
mile. On the Boston & Albany Railroad the expenditure 
assigned to "land and land damages'' averages over $25000 
per mile. Of course this includes some especially expensive 
land for terminals and stations in large cities. Less than $300 
per mile was assigned to this item by an unimportant 18-mile 
road. 

355. Item 4. CLEARING AND GRUBBING. The cost of this 
may vary from zero to 100% for miles at a time, but as an 
average figure it may be taken as about 3 acres per mile at a 
cost of say $50 per acre. The possibility of obtaining valuable 
timber, which may be utilized for trestles, ties, or otherwise, 
and the value of which may not only repay the cost of clearing 
and grubbing, but also some of the cost of the land, should not 
be forgotten. 

356. Item 5. Earthwork. This item also includes rock- 
work. The methods of estimating the cost of earthwork and 
rockwork have been discussed in Chapter III. The percentage 
of this item to the total cost is very variable. On a western 
prairie it might not be more than 5 to 10%. On a road through 
the mountains it T\ill run up to 20 or 25%, and even more. 
The item also includes tunneling, which on some roads is a 
heavy item. 

357. Item 6. BRIDGES, TRESTLES, AND CULVERTS. This item 
vnR usually amount to 5 or 6% of the total cost of the road. 
In special cases, where extensive trestling is necessary, or 
several large bridges are required, the percentage will be much 
higher. On the Other hand, a road whose route avoids the 
watercourses may have very little except minor culverts. On 
the Boston & Albany the cost is given as $5860 per mile; on 
the Adirondack Railroad, $2845 per mile. Considering their 
relative character (double and single track), these figures are 
^relatively what we might expect. 



430 



RAILROAD CONSTRUCTION. 



§ 358. 



358. Item 7. Trackwork. This item v/ill be considered as 
including everything above subgrade, except as otherwise 
itemized. 

(a) Ballast. With an average width, for single track, of 
10 feet and an average depth of 15 inches, 2444 cubic yards of 
bal«ast will be required. The Pennsylvania Railroad estimate is 
2500 yards of gravel per mile of single track. At an estimate 
of 60 c. per yard, this costs $1500 per mile. Broken-stone 
ballast must be filled out over the ends of the ties and there- 
fore more is required; 2800 cubic yards of broken stone at 
$1.25 per yard in place will cost $3500 per mile. 

(b) Ties. Ties cost an3rwhere from 80 c. down to 35 c. and 
even 25 c. At an average figure of 50 c, 2640 ties per mile 
will cost $1320 per mile of single track. The cheaper ties are 
usually smaller and more must be used per mile, and this, tends 
to compensate the difference in cost. 

The following tabular form is convenient for reference: 



TABLE XV.— NUMBER OF CROSS TIES PER MILE. 



Number per 


Average spacing 


Number 


33' rail. 


center to center. 


per mile. 


22 


18.0 inches 


3520 


21 


18.9 ' 




3360 


20 


19.8 * 




3200 


19 


20.9 • 




3040 


18 


22.0 * 




2880 


17 


23.3 * 




2720 


16 


24.75 * 




2560 


15 


26.4 * 




2400 


14 


28.3 * 




2240 


13 


30.5 •• 


2080 



(c) Rails. The total weight of the rails used per mile may 
best be seen by the tabular form. 

A convenient and useful rule to remember is that the number 
of long tons (2240 lbs.) per mile of single track equals the w^eight 
of the rail per yard times V". The rule is exact. For example, 
there are 3520 yards of rail in a mile of single track; at 70 lbs. 
per yard this equals 246400 lbs., or 110 long tons (exactly); 

but 70XV-==110- 

Any calculation of the required w^eight of rail for a given 
weight of rolling-stock necessarily depends on the assumptions 
w^hich are made regarding the support which the rails receive 
from the ties. This depends not only on the width and spacing 



§ 358. 



COST OF RAILROADS. 



431 



TABLE XVT. TONS PER MILE (wiTH COSt) OF RAILS OF 

VARIOUS WEIGHTS. 





Tons 








Tons 






Weight 


(22401b.) 


Cost at 


Cost at 


Weight 


(22401b.) 


Cost at 


Cost at 


in lbs. 


per mile 


$26 per 


$30 per 


in lbs. 


per mile 


$26 per 


$30 per 


• per yd. 


of single 
track. 


ton. 


ton. 


per yd. 


of single 
track. 


ton. 


ton. 


8 


12.571 


$326.86 


$377.14 


65 


102.143 


$2655.71 


$3064.29 


10 


15.714 


408.57 


4.71.43 


66 


103.714 


2696.57 


3111.43 


12 


18.857 


490.29 


565.71 


67 


105.286 


2737.43 


3158.59 


14 


22.000 


572.00 


660 . 00 


68 


106.857 


2778.29 


3205.79 


16 


25.143 


653.71 


754.20 


70 


110.000 


2860.00 


3300.00 


20 


31.429 


817.14 


942.86 


71 


111.571 


2900.86 


3347.14 


25 


39.286 


1021.43 


1178.57 


72 


113.143 


2941.71 


3394.29 


30 


47.143 


1225.71 


1414.29 


73 


114.714 


2982.57 


3441.43 


35 


55.000 


1430.00 


1650.00 


75 


117.857 


3064.29 


3535.71 


40 


62.857 


1634.29 


1885.71 


78 


122.571 


3186.86 


3677.14 


45 


70.714 


1838.57 


2121.43 


80 


125.714 


3268.57 


3771.43 


48 


75.429 


1961.14 


2262.86 


82 


128.857 


3350.29 


3865.71 


60 


78.571 


2042.86 


2357.14 


85 


133.571 


3472 . 86 


4007.14 


52 


81.714 


2124.57 


2451.43 


88 


138.286 


3595.43 


4148.57 


56 


88.000 


2288.00 


2640.00 


90 


141.429 


3677.14 


4242.86 


57 


89.571 


2328.86 


2687.14 


92 


144.571 


3758.86 


4337.14 


" 60 


94.286 


2451.43 


2828 . 57 


95 


149.286 


3881.43 


4478 . 57 


61 


95 . 857 


2492 . 29 


2875.71 


98 


154.000 


4004.00 


4620.00 


63 


99.000 


2574.00 


2970.00 


100 


157.143 


4085.71 


4714.29 



About two per cent. (2%) extra should be allowed for waste in cutting. 

of the ties (which are determinable), but also on the support 
which the ties receive from the ballast, which is not only very- 
uncertain but variable. No general rule can therefore claim 
any degree of precision, but the following is given by the Bald- 
win Locomotive Works: " Each ten pounds weight per yard of 
ordinary steel rail, properly supported by cross- ties (not less 
than 14 per 30-foot rail), is capable of sustaining a safe load 
per wheel of 2240 pounds." For example, a consolidation loco- 
motive with 112600 lbs. on 8 drivers has a load of 14075 lbs. 
per wheel. This divided b}^ 2240 gives 6.28. According to the 
rule, the rails for such a locomotive should weigh at least 62.8 
lbs. per 3^ard. 

(d) Splice-bars, track-bolts, and spikes. These are usually 
sold by the pound, except the patented forms of rail-joints, 
which are sold by the pair. In any case thej^ are subject to 
market fluctuations in price. As an approxim^ate value the 
following prices are quoted: Splice-bars, 1.35 c. per pound; 
track-bolts, 2.4 c; spikes, 1.75 c. The weight of the splice- 
bars will depend on the precise pattern adopted — its cross- 
section and length. • 



432 



RAILROAD CONSTRUCTION. 



358. 



In Table XVII are quoted from a catalogue of the Illinois 
Steel Co. the weights per foot of sections of angle-bars which 
they recommend for various weights of rail and which are de- 
signed to fit standard A. S. C. E. rail sections of those weights. 
The net weight of the angle-bars may be approximated by 
subtracting about 2.5% to 4% from the gross weight to allow 
for the bolt-holes. A deduction of 2.5% is usually about 
right for the heavier sections. Their recommendations regard- 
ing lengths of angle-bars do not include those for rails heavier, 
than 50 pounds per yard. On the basis of a length of 23 inches 
for four-hole splices and of 33 inches for six-hole splices, the 
weights of splice-bars have been computed for the several 
styles of splices for heavier rails, allowing 2.5% for the holes. 
The lengths recommended for track bolts are those which will 
allow about J inch for the nutlock and for margin, except for 
tbe lighter railsc 



TABLE XVII. — SPLICE-BARS FOR VARIOUS WEIGHTS OP RAILS. 



Weight 


Length 


Weight 


Weight 


Proper 


Proper size 


of 
rail. 


of 
angle-bar. 


per 
foot. 


of 
pair. 


size of 
track-bolt. 


of spikes. 


30 


21" 


4.49 


15.1 


2rxr 


4" X¥' 


35 


21" 


4.7 


15.9 


21" Xf" 


4rxr 


40 


21" 


5.54 


18.8 


3 "Xf" 


5 "xr 


45 


21" 


6.3 


21.5 


3 "Xf" 


5rXA" 


50 


21" 


6.97 


23.4 


3rxf" 


5rx^" 


55 


23" 


7.5 


28.0 


3rxi" 


5rxA" 


60 


23" 


8.4 


31.4 


3f"X|" 


5rxT^" 


65 


i23" 
\33" 


9.2 


34.4 


4 "Xf" 


5rxi^/' 


9.6 


51.5 


4i"xr 


5rxi%" 


70 


/23" 


9.0 


33.6 


4 "X-l" 


srx^" 


133" 


10.0 


53.6 


4 "Xf" 


5rxi^" 


15 


f23" 


10.68 


39.9 


4i"Xf" 


srxA" 


\33" 


11.9 


63.7 


4 "Xf" 
4i"Xr 


srxi^" 


80 


/23" 


10.61 


39.7 


5rxT%" 


\33" 


14.65 


78.5 


4rxf" 


5rxiV' 


85 


33" 


12.4 


66.4 


4y'xr 


5rx^"orr 


90 


33" 


13.5 


72.3 


4f"xr 

41" XF' 


5rX^"orr' 


95 


33" 


14.7 


78.7 


5rXi^/''orr 


100 


33" 


15.78 


85.0 


4f"XF' 


5r'Xt^"ort" 



358. 



COST OF RAILROADS. 



433 



TABLE XVIII. — RAILROAD SPIKES. 







Ties 24'' between cen- 






Average 


ters, 4 spi 


kes per tie, 


Suitable 


Size meas- 


number 


number 


per mile. 


weight of 


ured under 
head. 


per keg of 
200 pounds 






rail. 










Pounds. 


Kegs. 




5rxr' 


275 


7680 


38.40 


90 to 100 


5rXi^6" 


375 


5632 


28.16 


45 " 100 


5" XA" 


400 


5280 


26.40 


40 '• 56 


5- xr 


450 


4692 


23.46 


40 


AY'Xi" 


530 


3984 


19.92 


35 


4" xr 


600 


3520 


17.60 


30 


4rXi^" 


680 


3104 


15.52 


25 to 30 



TABLE XIX. TRACK-BOLTS. 

Average number in a keg of 200 pounds. 



Size of 


Square 


Hexagonal 


Suitable 


bolt. 


nut. 


nut. 


rail. 


3" X-" 


366 


395 


40 pound 


3„ ><i,/ 


250 


270 




3F'x-'' 


243 


261 




3¥'x-" 


236 


253 


50 


sr'x^" 


229 


244 


55 to 60 


4" X-" 


222 


236 


65 " 70 


4V'X-' 


215 


228 


75 


3rx-" 


170 


180 




3rxr 


165 


175 




4- xr 


161 


170 




4rxr 


157 


165 


80 


4rxr 


153 


160 


85 


4rxr 


149 


156 


90 



434 RAILROAD CONSTRUCTION. § 358. 

(e) Track-laying. Much depends on the force of men em- 
ployed and the use of systematic methods; $528 per mile is 
the estimate employed by the Pennsylvania Railroad. $500 per 
mile is the estimate given in § 362. 

359. Item 8. Buildings and Miscellaneous structures. 
Except for rough and preliminary estimates, these items must 
be individually estimated according to the circumstances. The 
subitems include depots, engine-houses, repair-shops, water- 
stations, section- and tool-houses, besides a large variety of 
smaller buildings. The structures include turn-tables, cattle- 
guards, fencing, road-crossings, overhead bridges, etc. The 
detailed estimate, given in § 362, illustrates the cost of these 
smaller items. 

360. Item 9. INTEREST ON CONSTRUCTION. The amount 
of capital that must be spent on a railroad before it has begun 
to earn anything is so very large that the interest on the cost 
during the period of construction is a very considerable item. The 
amount that must be charged to this head depends on the cur- 
rent rate of money on the time required for construction and 
on the abilit}^ of the capitalists to retain their capital where 
it will be earning something until it is actually needed to pay 
the company's obligations. Of course, it is not necessary to 
have the entire capital needed for construction on hand when 
construction commences. Assuming money to be worth 6%, 
that the work of construction will require one year, that the 
money may be retained where it will earn something for an 
average period of six months after construction commences, 
or, in other words, it will be out of circulation six months before 
the road is opened for traffic and begins to earn its way, then 
we may charge 3% on the total cost of construction. 

361. Item 10. Telegraph Lines. This evidently depends 
on the scale of the road and the magnitude of the business to 
be operated. In the following estimate it is given as $200 
per mile, which evidently is intended to apply to the business 
of a small road. 

362. Detailed estimate of the cost of a line of road. The fol- 
lowing estimate was given in the Engineering News of Dec. 27, 
1900, of the cost of the Duluth, St. Cloud, Glencoe & Mankato 
Railroad, 157.2 miles long. 

The estimate is exactly as copied from the Engineering Neivs. 
There are some numerical discrepancies. Item 26 should evi- 



§ 362c COST OF RAILROADS, 435 

dently be based on the sum of the first 25 items, and item 27 
on the smn of the first 26. The figures in parentheses ( ) are 
deduced from the figures given. 

1. Right-of-way: 1905.3 acres (12.12 acres per mile) @ $100 per 

acre $190530 

2. Clearing and grubbing. 144 acres (0.916 acre per mile) @ $50 

per acre 7200 

3. Earth excavation. 1907590 cu. yds. (12135 cu. yds. per mile) 

@ 15 c 286138 

4. Rock excavation. 5100 cu. yds. (32.44 cu. yds. per mile) @ 80 c. 4080 
( Wooden-box culverts . 508300 ft. B.M. @ $30 per M. . $15249 

i Iron-pipe culverts. 879840 lbs. @ 3c. per lb 26395 41644 

( PHe trestling; 4600 lin. ft. @ 35 c. per lin. ft 1610 

( Timber trestling. 509300 ft. B.M. @ $30 per M 15279 16889 

j Bridge masonry: 5520 cu. yds. @. $8 per cu. yd 44160 



i 



Bridges, iron, 100 spans, 2000000 lbs. @ 4 c. per lb. . . 80000 124160. 

8. Cattle-guards 8750 

9. Ties (2640 per mile) . 419813 (159.02 miles) (^ 35 c 146935 

10. Rails (70 lbs. per yd.): 110 tons per mile, 17492.2 tons (159.02 

miles @$26 384797 

11. Rail sidings (70 lbs. per yd.) : 110 tons per mile, 3300 tons 

(30 miles® $26 85800 

12. Switch timbers and ties 3300 

13. Spikes: 5920 lbs. per mile. 1107040 (187 m.) @ 1.75. c. per lb. 19373 

14. Splice-bars. 2635776 lbs. @, 1.35 c. per lb 35583 

15. Track-bolts (2 to joint (?)): 188458.3 lbs. @ 2.4 c. per lb 4520 

16. Track-laying 187.2 miles @, $500 per mile 93600 

17. BaUasting: 2152 cu. yds. per mile, 402854 (187.2 m.) @ 60 c. . 241712 

18. Turn-out and switch furnishings 6450 

19. Road-crossings, 68040 ft. B.M. (^ $30 per M 2041 

20. Section and tool-houses, 16 @ $800 12800 

21. Water-stations 15000 

22. Turn-tables, 6 @ $800 4800 

23. Depots, grounds, and repair-shops 78000 

24. Terminal grounds and special land damages 150000 

25. Fencing, 314 miles ($150 per mile) 47100 

26. Engineering and office expenses (5% of $1984458) 99222 

27. Interest on construction (3% of $2083680) 62510 

28. Rolling-stock ($5000 per mile) 786000 

29. Telegraph line: 157 miles @ $200 per mile 31400 

$3060340 
Average cost per mile ready for operation, $19467. 
Approximate cost of 130 miles from St. Cloud to Duluth, estimated at 

$23000 per mile. 
Approximate cost of entire line from Albert Lea to Duluth, 287.2 miles, 

$6050340 ($21060 per mile). 



PART II. 

EAILEOAD ECONOMICS. 



CHAPTER XVIII. 

INTRODUCTION. 

363. The magnitude of railroad business. The gross earnings 
of/ailroads for the year ending June 30, 1910, were $2,750,667,- 
435. This is greater than the combined value of all the gold, 
silver, iron, wheat, and corn produced by the country. 

About 1,700,000 persons (over one sixtieth of the population) 
were directly employed by the roads for a compensation of 
$1,143,725,306. Probably 5,000,000 to 6,000,000 people were 
supported by this. Besides all these, probably 8,000,000 
employes were kept busy in occupations which are a more or 
less direct result of railroads, e.g., locomotive- and car-shops, 
rail-mills, etc. We therefore may estimate that perhaps 25,000,- 
000 people (or over one fourth of our population) are supported 
by railroads or by occupations which owe their chief existence 
to railroads. 

The ^'number of passengers carried 1 mile^^ was 32,338 millions. 
Calling the population of the United States 92,000,000 for round 
numbers, it means an average ride of 351 miles for every man, 
woman, and child. 

The ^Hons carried 1 mile^^ were 255,017 millions, or nearly 
2773 ton-miles per inhabitant. The payments made to the 
railroads averaged nearly $30 per inhabitant. 

The actual bona-fide cost of the railroads of the country 
cannot be accurately computed (as will be shown later), but 
the capital, as represented by stocks and bonds, aggregated 

436 



§ 365. INTRODUCTION. 437 

in 1910, $18,417,132,238 or about $200 per inhabitant. This is 
roughly about one sixth of the total national wealth. 

The above figures may give some idea of the magnitude of 
the interests involved in the operation of railroads. No single 
business in the country approaches it in capital involved, earn- 
ings, number of people affected, or effect on other business. 

364. Cost of transportation. The importance of railroads 
may be also indicated by their power of creating cheap trans- 
portation. Less than one hundred years ago local famine 
and overabundant harvests within a radius of a few miles were 
not unknown. When the transportation of goods depended 
on actual porterage by human beings, as has been the case 
but recently in the Klondike, the transportation of 50 lbs. 20 
miles might be considered a hard day^s labor. At $2 per day, 
this equals $4 per ton-mile. In 1909 the railroads transported 
freight at an average cost to the pubHc of 0.763 c. per ton per 
mile, and the feeding of Europe with wheat from Manitoba 
has become a commercial possibiHty. In 1909 passengers paid 
an average charge of 1.928 c. per mile, and a trip of 1000 miles 
inside of 24 hours is now common. 

365. Study of railroad economics — its nature and limitations. 
The multiplicity of the elements involved in most problems 
in railroad construction preclude the possibihty of a solution 
which is demonstrably perfect. Barring out the compara- 
tively few cases in this country where it is difficult to obtain 
any practicable location, it may be said that a comparatively 
low order of talent will suffice to locate anywhere a railroad 
over which it is physically possible to run trains. It may be 
very badly located for obtaining business, the ruling grades 
may be excessive, the alignment may be very bad, and the 
road may be a hopeless financial failure, and yet trains can be 
run. Among the infinite number of possible locations of the 
road, the engineer must determine the route which will give 
the best railroad property for the least expenditure of money — ■ 
the road whose earning capacity is so great that after paying 
the operating expenses and interests on the bonds, the surplus 
available for di\ddends or improvements is a maximum. 

An unfortunate part of the problem is that even the blunders 
are not always readily apparent nor their magnitude. A de- 
fective dam or bridge will give way and every one realizes the 
failure, but a badly located railroad affects chiefly the finances 



438 EAILEOAD CONSTRUCTION. § 365, 

of the enterprise by a series of leaks which are only perceptible 
and demonstrable by an expert, and even he can only say that 
certain changes would probably have a certain financial value. 

366. Outline of the engineer's duties. The engineer must 
realize at the outset the nature and value of the conflicting 
interests which are involved in variable amount in each possi- 
ble route. 

(a) The maximum of business must be obtained, and yet it may 
happen that some of the business may only be obtained by an 
extravagant expenditure in building the line or by building a 
line very expensive to operate. 

(b) The ruling grades should be kept low, and yet this may 
require a sacrifice in business obtained and also may cost more 
than it is worth. 

(c) The alignment should be made as favorable as possible; 
favorable alignment reduces the future operating expenses, 
but it may require a very large immediate outlay. 

(d) The total cost must be kept within the amount at which 
the earnings will make it a profitable investment. 

(e) The road must be completed and operated until the 
"normal'' traffic is obtained and the road is self-supporting 
without exhausting the capital obtainable by the projectors ; 
for no matter how valuable the property may ultimately be- 
come, the projectors will lose nearly, if not quite, all they have 
invested if they lose control of the enterprise before it becomes 
a paying investment. 

Each new route suggested makes a new combination of the 
above conflicting elements. The engineer must select a route 
by first eliminating all lines which are manifestly impracticable 
and then gradually narrowing the choice to the best routes 
whose advantages are so nearly equal that a closer detailed 
comparison is necessary. 

The ruling grade and the details of alignment have a large 
influence on the operating expenses. A large part of this 
course of instruction therefore consists of a study of operating 
expenses under average normal conditions, and then a study 
of the effect on operating expenses of given changes in the align- 
ment. 

367, Justification of such methods of computation. It may 
be argued that the data on which these computations are based 
are so unreliable (because variable and to some extent non- 



§ 367, INTRODUCTION. * 439 

computable) that no dependence can be based on the conclu- 
sions. This is true to the extent that it is useless to claim 
great precision in the computation of the value of any pro- 
posed change of alignment. Suppose, for example, it is com- 
puted that a given improvement in alignment will reduce the 
operating expenses of 20 trains per day by $1000 per year. 
Suppose the change in alignment may be made for $5000, which 
may be obtained at 5% interest. Even with large allowances 
for inaccuracy in the computation of the value, $1000, it evi- 
dently will be better to incur an additional interest charge of 
$250 than increase the annual operating expenses by $1000. 
Moreover, since traffic is almost sure to increase (and interest 
charges are generally decreasing), the advantage of the im- 
provement will only increase as time passes. On the other 
hand, if the improvement cannot be made except by an expen- 
diture of, say, $50000, the change would evidently be unjus- 
tifiable. When the interest on the first cost is practically 
equal to the annual operating value of the proposed improve- 
ment, there is evidently but little choice; no great harm can 
result from either decision, and the decision frequently will 
depend on the willingness to increase the total amount invested 
in the enterprise. 

To express the above question more generally, in every com- 
putation of the operating value of a proposed improvement, 
it may always be shown that the true value lies somewhere 
between some maximum and some minimum. Closer calcula- 
tions and more reliable data will narrow the range between 
these extreixie values. According as the interest on the cost 
of the proposed improvement is greater or less than the mean 
of these limits, we may judge of its advisability. The range 
of the limits shows the uncertainty. If it lies outside of the 
limits there is no uncertainty, assuming that the limits have 
been properly determined. If well within the limits, either 
decision will answer unless other considerations determine the 
question. And so, although it is not often possible to obtain 
precise values, we may generally reach a conclusion which is 
unquestionable. Even under the most unfavorable circum- 
stances, the computations, when made with the assistance of 
all the broad common sense and experience that can be brought 
to bear, will point to a decision which is much better than mere 
''judgment,'' which is responsible for very many glaring and 



440 RAILROAD CONSTRUCTION. § 367. 

costly railroad blunders. In short, Railroad Economics means 
the application of systematic methods of work plus experience 
and judgment, rather than a dependence on judgment unsys- 
tematically formed. It makes no pretense to furnishing mechan- 
ical rules by which all railroad problems may be solved by any 
one, but it does give a general method of applying principles 
by which an engineer of experience and judgment can apply 
his knowledge to better advantage. To the engineer of limited 
experience the methods are invaluable; without such methods 
of work his opinions are practically worthless; with them 
his conclusions are frequently more sound than the unsystem- 
atically formed judgments of a man with a glittering record. 
But the engineer of great experience may use these methods 
to form the best opinions which are obtainable, for^ he can apply 
his experience to make any necessary local modifications in the 
method of solution. The dangers lie in the extremes, either 
recklessly applying a rule on the basis of insufficient data to 
an unwarrantable extent, or, disgusted with such evident 
unreliability, neglecting altogether such systematic methods of 
work. 



CHAPTER XIX. 

THE PROMOTION OF RAILROAD PROJECTS. 

368. Method of formation of railroad corporations. Many 

business enterprises, especially the smaller ones, are financed 
entirely by the use of money which is put into them directly 
in the form of stock or mere partnership interest. A railroad 
enterprise is frequently floated vnth. a comparatively small 
financial expenditure on the part of the original promoters. 
The promoters become convinced that a railroad between A 
and Bj passing through the intermediate towns of C and D, 
with others of less importance, will be a paying investment. 
They organize a company, have surveys made, obtain a charter, 
and then, being still better able (on account of the additional 
information obtained) to exploit the financial advantages of 
their scheme, they issue a prospectus and invite subscriptions 
to bonds. Sometimes a portion of these bonds are guaranteed, 
principal and interest, or perhaps the principal alone, by town- 
ships or by the national government. The cost of this pre- 
liminary work, although large in gross amount if the road is 
extensive, is yet but an insignificant proportion of the total 
amount involved. The proportionate amount that can be 
raised by means of bonds varies vnth. the circumstances. In 
the early history of railroad building, when a road was pro- 
jected into a new country where the trafiic possibilities were 
great and there was absolutely no competition, the financial 
success of the enterprise would seem so assured that no diffi- 
culty would be experienced in raising from the sale of bonds 
all the money necessary to construct and equip the road. But 
the promoters (or stockholders) must furnish all money for the 
preliminary expenses, and must make up all deficiencies be- 
tween the proceeds of the sale of the bonds and the capital needed 
for construction. 

^^In theory, stocks represent the property of the responsible 
owners of the road, and bonds are an encumbrance on that 

441 



442 RAILROAD CONSTRUCTION. § 368. 

property. According to this theory, a railroad enterprise 
should begin with an issue of stock somewhere near the value 
of the property to be created and no more bonds should be 
issued than are absolutely necessary to complete the enter- 
prise. Now it is not denied that there are instances in which 
this theory is followed out. In New England, for example, 
as well as in some of the Southern States, there are a few roads 
represented wholly by stock or very lightly mortgaged. But 
this theory does not conform to the general history of railway 
construction in the United States, nor is it supported by the 
figures that appear in the summary. The truth is, railroads 
are built on borrowed capital, and the amount of stock that is 
issued represents in the majority of cases the difference between 
the actual cost of the undertaking and the confidence of the 
public expressed by the amount of bonds it is willing to absorb 
in the ultimate success of the venture." * 

''The same general law obtains and has always obtained 
throughout the world, that such properties (as railways) are 
always built on borrowed money up to the limit of what is 
regarded as the positive and certain minimum value. The 
risk only — the dubious margin which is dependent upon sagac- 
ity, skill, and good management — is assumed and held by the 
company proper who control and manage the property." f 

369. The two classes of financial interests — the security and 
profits of each. From the above it may be seen that stocks, 
bonds, car-trust obligations, and even current liabilities repre- 
sent railroad capital. The issue of the bonds ''was one means 
of collecting the capital necessary to create the property against 
which the mortgage lies." The variation between these inter- 
ests lies chiefly in the security and profits of each. The current 
liabilities are either discharged or, as frequently happens, they 
accumulate until they are funded and thus become a definite 
part of the railroad capital. 

The growth of this tendency is shown in the following tabular 
form (see next page) : 

The bonded interest has greater security than the stock, but 
less profit. The interest on the bonds must be paid before any 
money can be disbursed as dividends. If the bond interest 



* Henry C. Adams, Statistician, U. S. Int. Con. Commission. 
t A. M. Wellington, Economic Theory of Railway Location. 



§369. 



PROMOTION OF RAILROAD PROJECTS. 



443 



Capitalization of 


June 30. 1888. 


June 30. 1898. 


June 30. 1910. 


Railroads in the United 
States. 


Amount, 
millions. 


Per 

cent. 


Amount, 
millions. 


Per 

cent. 


Amount, 
millions. 


Per 
cent. 


Stocks 


3864 

3869 

396 


47.5 

47.6 

4.9 


5311 
5510 
1087 


44.6 
46.31 
9.1/ 


8,113 
10,303 


44.05 


Funded debt , . 


Current liabilities, etc. . 


55.95 



is not paid, a receivership, and perhaps a foreclosure and sale 
of the road, is a probability, and in such case the stockholder's 
interests are frequently wiped out altogether. The bond- 
holder's real profit is frequently very different from his nomi- 
nal profit. He sometimes buys the bonds at a very considerable 
discount, which modifies the rate which the interest received 
bears to the amount really invested. Even the bondholder's 
security may suffer if his mortgage is a second (or fifth) mort- 
gage, and the foreclosure sale fails to net sufficient to satisfy 
all previous claims. 

On the other hand, the stockholder, who may have paid in 
but a small proportion of his subscription, mayy if the venture 
is successful, receive a dividend which equals 50 or 100% of the 
money actually paid in, or, as before stated, his entire holdings 
may be entirely wiped out by a foreclosm-e sale. When the 
road is a great success and the dividends very large, additional 
issues of stock are generally made, which are distributed to the 
stockholders in proportion to their holdings, either gratuitously 
or at rates which give the stockholders a large advantage over 
outsiders. This is the process known as ^^ watering." While 
it may sometimes be considered as a legitimate '^salting down" 
of profits, it is frequently a cover for dishonest manipulation of 
the money market. 

For the twelve years between 1887 and 1899 about two thirds 
of all the railroad stock in the United States paid no dividends, 
while of those that paid dividends the average rate varied 
from 4.96 to 5.74%. The year from June 30, 1898, to June 30, 
1899, was the most prosperous year of the group, and yet nearly 
60% of all railroad stock paid no dividend, and the average 
rate paid by those which paid at all was 4.96%. The total 
amount distributed in dividends was greater than ever before, 
but the average rate is the least of the above group because many 
roads, which had passed their dividends for many previous 



444 RAILROAD CONSTRUCTION. § 369, 

years, distinguished themselves by declaring a dividend, even 
though small. During that same period but 13.35% of the 
stock paid over 6% interest. The total dividends paid amounted 
to but 2.01% of all the capital stock, while investments ordi- 
narily are expected to yield from 4 to 6% (or more) according 
to the risk. Of course the effect of ^^ watering'' stock is to 
decrease the nominal rate of dividends, but there is no dodging 
the fact that, watered or not, even in that year of ^'good times," 
about 60% of all the stock paid no dividends. Unfortunately 
there are no accurate statistics showing how much of the stock 
of railroads represents actual paid-in capital and how much 
is ''water." The great complication of railroad finances and 
the dishonest manipulation to which the finances of some rail- 
roads have been subjected would render such a computation 
practically worthless and hopelessly unreliable now. 

During the year ending June 30, 1898 (which may in general 
be considered as a sample), 15.82% of the funded debt paid no 
interest. About one third of the funded debt paid between 
4 and 5% interest, which is about the average which is paid. 

The income from railroads (both interest on bonds and divi- 
dends on stock) may be shown graphically by diagrams, such 
as are given in the annual reports of the Interstate Commerce 
Commission. They show that while railroad investments are 
occasionally very profitable, the average return is less than 
that of ordinary investments to the investors. The indirect 
value of railroads in building up a section of country is almost 
incalculable and is worth many times the cost of the roads. 
It is a discouraging fact that very few railroads (old enough to 
have a history) have escaped the experience of a receivership, 
with the usual financial loss to the then stockholders. But 
there is probably not a railroad in existence which, however 
much a financial failure in itself, has not profited the community 
more than its cost. 

370. The small margin between profit and loss to projectors. 
When a railroad is built entirely from the funds furnished by 
its promoters (or from the sale of stock) it will generally be a 
paying investment, although the rate of payment may be very 
small. The percentage of receipts that is demanded for actual 
operating expenses is usually about 67%. The remainder will 
usually pay a reasonable interest on the total capital involved. 
But the operating expenses are frequently 90 and even 100% of 



§ 371. PROMOTION OF RAILROAD PROJECTS. 445 

the gross receipts. In such cases even the bondholders do not 
get their due and the stockholders have absolutely nothing. 
Therefore the stockholder's interest is very speculative. A 
comparatively small change in the business done (as is illus- 
trated numerically in § 372) will not only wipe out altogether the 
dividend — taken from the last small percentage of the total 
receipts and which may equal 50% or more of the capital stock 
actually paid in — but it may even endanger the bondholders' 
security and cause them to foreclose their mortgage. In such 
a case the stockholders' interest is usually entirely lost. It 
does not alter the essential character of the above-stated rela- 
tions that the stockholders sometimes protect themselves 
somewhat by buying bonds. By so doing they simply decrease 
their risk and also decrease the possible profit that might result 
from the investment of a given total amount of capital. 

371. Extent to which a railroad is a monopoly. It is a popu- 
lar fallacy that a railroad, when not subject to the direct com- 
petition of another road, has an absolute monopoly — that it 
controls ^^all the traffic there is" and that its income will be 
practically independent of the facilities afforded to the public. 
The growth of railroad traffic, like the use of the so-called 
necessities or luxuries of life, depends entirely on the supply 
and the cost (in money or effort) to obtain it. A large part of 
railroad traffic belongs to the unnecessary class — such as travel- 
ing for pleasure. Such traffic is very largely affected by mere 
matters of convenience, such as well-built stations, -convenient 
terminals, smooth track, etc. The freight traffic is very largely 
dependent on the possibility of delivering manufactured articles 
or produce at the markets so that the total cost of production 
and transportation shall not exceed the total cost in that 
same market of similar articles obtained elsewhere. The crea- 
tion of facilities so that a factory or mine may successfully 
compete with other factories or mines will develop such traffic. 
The receipts from such a traffic may render it possible to still 
further develop facilities which will in return encourage further 
business. On the other hand, even the partial withdrawal of 
such facilities may render it impossible for the factory or mine 
to compete successfully with rivals; the traffic furnished by 
them is completely cut off and the railroad (and indirectly the 
whole community) suffers correspondingly. The ^^ strictly 
necessary" traffic is thus so small that few railroads could pay 



446 



RAILROAD CONSTRUCTION. 



§372. 



their operating expenses from it. The dividends of a road 
come from the last comparatively small percentage of its revenue, 
and such revenue comes from the '^ unnecessary" traffic which 
must be coaxed and which is so easily affected by apparently 
insignificant ^ ' conveniences . ' ' 

372. Profit resulting from an increase in business done; loss 
resulting from a decrease. In a subsequent chapter it will 
be shown that a large portion of the operating expenses are 
independent of small fluctuations in the business done and that 
the operating expenses are roughly two thirds of the gross 
revenue. Assume that by changes in the alignment the business 
obtained has been increased (or diminished) 10%. Assume for 
simplicity that the operating expenses on the revised track 
are the same as on the route originally planned; also that the 
cost of the track is the same and hence the fixed charges are 
assumed to be constant for all the cases considered. Assume 
the fixed charges to be 28%. The additional business, when 
carried in cars otherwise but partly filled will hardly increase 
the operating expenses by a measurable amount. When 
extra cars or extra trains are required, the cost will increase 
up to about 60% of the average cost per train mile. We may 
say that 10% increase may in general be carried at a rate of 
40% of the average cost of the traffic. A reduction of 10% 
in traffic may be assumed to reduce expenses a similar amount. 
The effect of the change in business will therefore be as follows: 





Business increased 10%. 


Business decreased 10%. 


Operating exp. = 67 
Fixed charges = 28 


67(1 + 10%X40%)= 69.68 
28.00 


67(1 - 10% X 40%) = 


= 64.32 
. 28 . 00 




Income 

Deficit 




95 
Total income. . . 100 


97.68 
Income 110.00 


92.32 
. 90.00 


Available for divi- 
dends 5 


Available for divi- 
dends 12 . 32 


. 2.32 



In the one case the increase in business, which may often 
be obtained by judicious changes in the alignment or even by. 
better management without changing the alignment, more than 
doubles the amount available for dividends. In the other case 
the profits are gone, and there is an absolute deficit. The 
above is a numerical illustration of the argument, previously 



§ 373, PROMOTION OF RAILROAD PROJECTS. 447 

stated, of the small margin between profit and loss to the original 
projectors. 

373. Estimation of probable volume of traffic and of probable 
growth. Since traffic and traffic facilities are mutually inter- 
dependent and since a large part of the normal traffic is merely 
potential until the road is built, it follows that the traffic of a 
road will not attain its normal volume until a considerable 
time after it is opened for operation. But the estimation even 
of this normal volume is a very uncertain problem. The esti- 
mate may be approached in three ways: 

1st. The actual gross revenue derived by all the railroads 
in that section of the country (as determined by State or U. S. 
Gov. reports) may be divided by the total population of the 
section and thus the average annual expenditure per head of 
population may be determined. A determination of this value 
for each one of a series of years will give an idea of the normal 
rate of growi:h of the traffic. Multiplying this annual contri- 
bution by the population which may be considered as tributary- 
gives a valuation of the possible traffic. Such an estimate is 
unreliable (a) because the average annual contribution may not 
fit that particular locality, (Jb) because it is very difficult to 
correctly estimate the number of the true tributary population 
especially w^hen other railroads encroach more or less into the 
territory. Since a rough value of this sort may be readily 
determined, it has its value as a check, if for nothing else. 

2d. The actual revenue obtained by some road whose 
circumstances are as nearly as possible identical with the road 
to be considered may be computed. The weak point consists 
in the assumption that the character of the two roads is identical 
or in incorrectly estimating the allowance to be made for ob- 
served differences. The method of course has its value as a 
check. 

3d. A laborious calculation may be made from an actual 
study of the route — determining the possible output of all 
factories, mines, etc., the amount of farm produce and of lumber 
that might be shipped, with an estimate of probable passenger 
traffic based on that of like towns similarly situated. This 
method is the best when it is properly done, but there is always 
the danger of lea^dng out sources of income — both existent 
and that to be developed by traffic facilities, or, on the other 
handj of overestimating the value of expected traffic. In the 



448 



RAILROAD CONSTRUCTION. 



§ 373. 



following tabular form are shown the population, gross re- 
ceipts, receipts per head of population, mileage, earnings per 
mile of line operated, and mileage per 10,000 of population for 
the whole United States. It should be noted that the values 
are only averages, that individual variations are large, and that 
only a very rough dependence may be placed on them as applied 
to any particular case. 



Year. 


Population 
(estimated) . 


Gross 
receipts. 


Receipts 
per head 
of popu- 
lation. 


Mileaget 


Earnings 

per mile 

of line 

operated. 


Mileage 
per 
10,000 
popula- 
tion, t 


1888... 
1889... 
1890... 
1891... 


60,100,000 

61,450,000 

*62,801,571 

64,150,000 


$910,621,220 

964,816,129 

1051,877,632 

1096,761,395 


$15.15 
15.81 
16.75 
17.10 


136,884 
153,385 
156,404 
161,275 


$6653 
6290 
6725 
6801 


24.94 
25.67 
26.05 
26.28 


1892... 
1893... 
1894... 
1895... 
1896... 


65,500,000 
68,850,000 
68,200,000 
69,550,000 
70,900,000 


1171,407,343 
1220,751,874 
1073,361,797 
1075,371,462 
1150,169,376 


17.89 
18.26 
15.74 
15.46 
16.22 


162,397 
169,780 
175,691 
177,746 
181.983 


7213 
7190 
6109 
6050 
6320 


26.19 
26.40 
26.20 
25.97 
25.78 


1897... 
1898... 
1899... 
1900.. . 
1901... 


72,350,000 
73,600,000 
74,950,000 
*76,295,220 
77,863,000 


1122,089,773 
1247,325,621 
1313,610,118 
1487,044,814 
1588,526,037 


15.53 
16.95 
17.53 
19.49 
20.47 


183,284 
184,648 
187,535 
192,556 
195,562 


6122 
6755 
7005 
-7722 
8123 


25.53 
25.32 
25.25 
25.44 
25.52 


1902... 
1903... 
1904. . . 
1905... 
1906. . . 


79,431,000 
80,998,000 
82,566,000 
84,134,000 
85,701,000 


1726,380,267 
1900,846,907 
1975,174,091' 
2082,482,406 
2325,765,167 


21.88 
23.70 
24.23 
25.15 
27.65 


200,155 
205,314 
212,243 
216,974 
222,340 


8625 
9258 
9306 
9508 
10460 


25.76 
26.03 
26.34 
26.44 
26.78 


1907. . . 
1908... 
1909... 
1910... 


87,279,000 

88,837,000 

90,405,000 

*91,972,266 


2589,105,578 
2393,805,989 
2418,677,538 
2750,667,435 


29.63 
26.95 
26.71 
29.91 


227,455 
231,540 
234,800 
238,609 


11383 
10338 
10301 
11528 [ 


26.38 
26.30 
26.20 
26.14 



* Actual. 



t Excludes a small percentage not reporting "gross receipts." 
t Actual mileage. 



The probable growth in traffic, after the traffic has once 
attained its normal volume, is a small but almost certain quantity. 
In the above tabular form this is indicated by the gradual 
growth in '^ receipts per head of population'^ from 1897 to 
1907. Then the sudden drop due to the panic of 1907 is clearly 
indicated, and also the gradual growth in the last few years. 
Even in England, where the population has been nearly station- 
ary for many years, the growth though small is unmistakable. 
On the other hand the growth in some of the Western States 



§ 37i. PROMOTION OF RAILROAD PROJECTS. 449 

has been very large. For example, the gross earnings per head 
of population in the State of Iowa increased from $1.42 in 1862 
to $10.00 in 1870, and to $19.46 in 1884. 

There will seldom be any justification in building to accommo- 
date a larger business than what is ''in sight." Even if it 
could be anticipated with certainty that a large increase in 
business would come in ten years, there are many reasons why 
it would be unwise to build on a scale larger than that required 
for the business to be immediately handled. Even though it 
may cost more in the future to provide the added accommo- 
dations {e.g. larger terminals, engine-houses, etc.), the extra 
expense will be nearly if not quite offset by the interest saved 
by avoiding the larger outlay for a period of years which may 
often prove much longer than was expected. A still more im- 
portant reason is the avoidance of uselessly sinking money at 
a time when every cent may be needed to insure the success 
of the enterprise as a whole. 

374. Probable number of trains per day. Increase with 
growth of traffic. The number of passenger trains per day 
cannot be determined by dividing the total number of passengers 
estimated to be carried per day by the capacity of the cars 
that can be hauled by one engine. There are many small 
railroads, running three or four passenger trains per day each 
way, which do not carry as many passengers all told as are 
carried on one heavy train of a trunk line. But because the 
bulk of the passenger traffic, especially on such light-traffic 
roads, is " unnecessary" traffic (see § 371) and must be encouraged 
and coaxed, the trains must be run much more frequently 
than mere capacity requires. The minimum number of passen- 
ger trains per day on even the lightest-traffic road should be 
two. These need not necessarily be passenger trains exclusively. 
They may be mixed trains. 

The number required for freight service may be kept more 
nearly according to the actual tonnage to be moved. At least 
one local freight will be required, and this is apt to be considerably 
Tvdthin the capacity of the engine. Some very light-traffic 
roads have little else than local freight to handle, and on such 
there is less chance of economical management. Roads ^dth 
heavy traffic can load up each engine quite accurately according 
to its hauling capacity and the resulting economy is great. Fluc- 
tuations in traffic are readily allowed for by adding on or drop- 



450 RAILROAD CONSTRUCTION, § 374. 

ping off one or more trains. Passenger trains must be run on 
regular schedule, full or empty. Freight trains are run by 
train-despatcher's orders. A few freight trains per day may be 
run on a nominal schedule, but all others will be run as extras. 
The criterion for an increase in the number of passenger trains 
is impossible to define by set rules. Since it should always 
come before it is absolutely demanded by the train capacity 
being overtaxed, it may be said in general terms that a train 
should be added when it is believed that the consequent in- 
crease in facilities will cause an increase in traffic the value of 
which will equal or exceed the added expense of the extra train. 
. 375. Effect on traffic of an increase in facilities. The term 
facilities here includes everything which facilitates the transport 
of articles from the door of the producer to the door of the 
consumer. As pointed out before, in many cases of freight 
transport, the reduction of facilities below a certain point will 
mean the entire loss of such traffic owing to local inability to 
successfully compete with more favored localities. Sometimes 
owing to a lack of facilities a railroad company feels compelled 
to pay the cartage or to make a corresponding reduction on 
what would normally be the freight rate. In competitive freight 
business such a method of procedure is a virtual necessity in 
order to retain even a respectable share of the business. Even 
though the railroad has no direct competitor, it must if possible 
enable its customers to meet their competitors on even terms. 
In passenger business the effect of facilities is perhaps even 
more marked. The pleasure travel will be largely cut down 
if not destroyed. 

376. Loss caused by inconvenient terminals and by stations 
far removed from business centers. This is but a special case 
of the subject discussed just in the preceding paragraph. The 
competition once existing between the West Shore and the 
New York Central was hopeless for the West Shore from the 
start. The possession of a terminal at the Grand Central 
Station gave the New York Central an advantage over the West 
Shore with its inconvenient terminal at Weehawken which 
could not be compensated by any obtainable advantage by 
the West Shore. This is especially true of the passenger busi- 
ness. The through freight business passing through or termi- 
nating at New York is handled so generally by means of floats 
that the disadvantage in this respect is not so great. The 



§ 376. PROMOTION OF RAILROAD PROJECTS. 451 

enormous expenditure (roughly $10,000,000) made by the 
Pennsylvania R. R., on the Broad Street Station (and its ap- 
proaches) in Philadelphia, a large part of which was made in 
crossing the Schuylkill River and running to City Hall Square, 
rather than retain their terminal in West Philadelphia, is an 
illustration of the policy of a great road on such a question. 
The fact that the original plan and expenditure has been very 
largely increased since the first construction proves that the 
management has not only approved the original large outlay, 
but saw the wisdom of making a very large increase in the ex- 
penditure. 

The construction of great terminals is comparatively infrequent 
and seldom concerns the majority of engineers. But an engineer 
has frequently to consider the question of the location of a 
way station with reference to the business center of the towTi. 
The following points may (or may not) have to be considered, 
and the real question consists in striking a proper balance 
between conflicting considerations. 

(1) During the early history of a railroad enterprise it is 
especially needful to avoid or at least postpone all expenditures 
which are not demonstrably justifiable. 

(2) The ideal place for a railroad station is a location im- 
mediately contiguous to the business center of the town. The 
location of the station even one fourth of a mile from this may 
result in a loss of business. Increase this distance to one mile 
and the loss is very serious. Increase it to five miles and the 
loss approaches 100%. 

(3) The cost of the ideal location and the necessary right 
of way may be a very large sum of money for the new enterprise. 
On the other hand the increase in property values and in the 
general prosperity of the towTi, caused by the railroad itself, 
will so enhance the value of a more convenient location that its 
cost at some future time will generally be extravagant if not 
absolutely prohibitory. The original location is therefore under 
ordinary conditions a finality. 

(4) To some extent the railroad will cause a movement of 
the business center toward it, especially in the establishment 
of new business, factories, etc., but the disadvantages caused 
to business already established is permanent. 

(5) In any attempt to compute the loss resulting from a 
location at a given distance from the business center it must be 



452 RAILROAD CONSTRUCTION. § 376. 

recognized that each problem is distinct in itself and that any 
change or growth in the business of the town changes the amount 
of this loss. 

The argument for locating the station at some distance from 
the center of the town may be based on (a) the cost of right 
of way, thus involving the question of a large initial outlay, 
(b) the cost of very expensive construction (e.g, bridges), 
again involving a large initial outlay, (c) the avoidance of ex- 
cessive grade into and out of the town. It sometimes happens 
that a railroad is following a line which would naturall}^ cause 
it to pass at a considerable elevation above (rarely below) 
the town. In this case there is to be considered not only the 
possible greater initial cost, but the even more important* increase 
in operating cost due to the introduction of a very heavy grade. 
To study such a case, compute the annual increase in operating 
expenses due to the additional grade, curvature, and distance; 
add to this the annual interest on the increased initial cost 
(if any) and compare this sum with the estimated annual loss 
due to the inconvenient location. The estimation of the increase 
in operating expenses is discussed in a subsequent chapter. 
The loss of business due to inconvenient location can only be 
guessed at. Wellington says that at a distance of one mile 
the loss would average 25%, with upper and lower limits of 
10 and 40%, depending on the keenness of the competition 
and other modifying circumstances. For each additional mile 
reduce 25% of the preceding value. While such estimates are 
grossly approximate, yet with the aid of sound judgment they 
are better than nothing and may be used to check gross errors. 

377. General principles which should govern the expenditure 
of money for railroad purposes. It will be shown later that 
the elimination of grade, curvature, and distance have a positive 
money value ; that the reduction of ruling grade is of far greater 
value ; that the creation of facilities for the handling of a large 
traffic is of the highest importance and yet the added cost of 
these improvements is sometimes a large percentage of the 
cost of some road over which it would be physically possible 
to run trains between the termini. 

The subsequent chapters will be largely devoted to a discussion 
of the value of these details, but the general principles governing 
the expenditure of money for such purposes may be stated as 
follows: 



§ 377. PROMOTION OF RAILROAD PROJECTS. 453 

1. No money should be spent (bej^ond the unavoidable 
minimum) unless it may be shown that the addition is in itself 
a profitable investment. The additional sum may not wreck 
the enterprise and it may add something to the value of the 
road, but unless it adds more than the improvement costs it is 
not justifiable. 

2. If it may be positively demonstrated that an improvement 
will be more valuable to the road than its cost, it should certainly 
be made even if the required capital is obtained with difficulty. 
This is all the more necessary if the neglect to do so will per- 
manently hamper the road with an operating disadvantage 
which will only grow worse as the traffic increases. 

3. This last principle has two exceptions: (a) the cost of 
the improvement may wreck the whole enterprise and cause 
a total loss to the original investors. For, unless the original 
promoters can build the road and operate it imtil its stock 
has a market value and the road is beyond immediate danger 
of a receivership, they are apt to lose the most if not all of 
their investment; (h) an improvement which is very costly 
although unquestionably wise may often be postponed by means 
of a cheap temporary construction. Cases in point are found 
at many of the changes of alignment of the Pennsylvania R. R., 
the N. Y., N. H. & H. R. R., and many others. While some of 
the cases indicate faulty original construction, at many of the 
places the original construction was wise, considering the then 
scanty traffic, and now the improvement is wise considering 
the great traffic. 



CHAPTER XX. 

OPERATING EXPENSES. 

378. Distribution of gross revenue. When a railroad com- 
prises but one single property, owned and operated by itself, 
the distribution of the gross revenue is a comparatively simple 
matter. The operating expenses then absorb about two thirds 
of the gross revenue; the fixed charges (chiefly the interest on 
the bonds) require about 25 or 30% more, leaving perhaps 3 
to 8% (more or less) available for dividends. The report on 
the Fitchburg R. R. for 1898 shows the following: 

Operating expenses $5,083,571 69. 1% 

Fixed charges 1,567,640 21 .3% 

Available for dividends, surplus, or per- 
manent improvements 708,259 9.6% 

Total revenue $7,359,470 100 . 0% 

But the financial statements of a large majority of the railroad 
corporations are by no means so simple. The great consolida- 
tions and reorganizations of recent years have been effected 
by an exceedingly complicated system of leases and sub-leases, 
purchases, "mergers,'' etc., whose forms are various. Railroads 
in their corporate capacity frequently own stocks and bonds 
of other corporations (railroad properties and otherwise) and 
receive, as part of their income, the dividends (or bond interest) 
from the investments. 

In . consequence of this complication, the U. S. Interstate 
Commerce Commission presents a ^'condensed income account'' 
of which the following is a sample (1910): 

Rail operations, operating revenues $2,750,667,435 

expenses 1,822,630,433 

Net operating revenue $ 928,037,002 

Net revenue from outside operations 2,225,455 

Total net revenue 930,262,457 

Taxes accrued 98,034,593 

Operating income 832,227,864 

Other income (chiefly dividends and interest on 

stocks and bonds owned) 252,219,946 

Gross corporate income $1,084,447,810 

454 



§ 378. OPEEATING EXPENSES. 455 

Deductions from gross corporate income (chiefiy in 
terest on funded debt, rents for lease of other 
roads and hire of equipment) 567,853,088 

Net corporate income , . . $ 516,594,722 

Dividends ^283,411,828 

Appropriations for additions, better- 
ments, new lines, and extensions . 55,061,675 
Appropriations for other reserves .... 2,640,893 

Total 1341,114,396 

Balance to credit of profit and loss 175,480,326 

In the above account an item of income (e.g., lease of road) 
reported by one road will be reported as a '^ deduction from 
income" by the road which leases the other. 

The above statement may be reduced to an income account 
of all the railways considered as one system. We then have 

Operating revenues 52,750,667,435 

Clear income from investments 78,442,027 



2,829,109,462 

Operating expenses $1,822,630,433 

Salaries and maintenance of leased 

lines 332,242 



Total 1,822,962,675 



Net revenues and income 1,006,146,787 

Net interest on funded debt 370,092,222 

Other interest 16,520,342 

Taxes 103,795,701 



Total 490,408,265 



Available for dividends, adjustments and improve- 
ments 515,738,522 

Net dividends 293,836,863 



Available for adjustments and improvements 221,901,659 

Of the balance ''available for adjustments and improvements," 
part was spent in permanent improvements, part was advanced 
to cover deficits in the operation of weak lines, and more than 
half was left as ''surplus," i.e., working capital. 

The percentages of the gross revenue which are devoted to 
operating expenses, fixed charges, and dividends are not neces- 
sarily an indication of creditable management or the reverse. 
Causes utterly beyond the control of the management, such 
as the local price of coal, may abnormally increase certain 



456 



EAILROAD CONSTRUCTION. 



§ 378. 



items of expense, while ruinous competition may cut down the 
gross revenue so that httle or nothing is left for dividends. 
A favorable location will sometimes make a road prosperous 
in spite of bad management. On the other hand, the highest 
grade of skill will fail to keep some roads out of the hands of 
a receiver. 

379. Fivefold distribution of operating expenses. The dis- 
tribution of operating expenses here used is copied from the 
method of the Interstate Commerce Commission. The aim is to 
divide the expenses into groups which are as mutually indepen- 
dent and distinct as possible — although, as will be seen later, 
a change in one item of expense will variously affect other 
items. 



Operating expenses. 

Maintenance of way and structures 

Maintenance of equipment 

Traffic expenses 

Transportation expenses 

General expenses 



1908. 



1909. 



1910. 



19.739 
22.06 

2.89 
52.01 

3.13 



19.29% 
22.75 

3.08 
50.90 

3.98 



100.00 



100.00 



20.22% 
22.66 

3.07 
50.29 

3.76 



100.00 



The above percentages represent the averages given by the 
reports for the three years from 1908 to 1910 inclusive. 

380. Operating expenses per train-mile. The reports of the 
U. S. Interstate Commerce Commission give the average cost 
per train-mile for every railroad in the United States. Although 
there are wide variations in these values, it is remarkable that 
the very large majority of roads give values which agree to 
within a small range, and that within this range are found not 
only the great trunk lines with their enormous train mileage, 
but also roads with very light traffic. 

In the following tabular form is shown a statement taken from 
the reports for 1904 and for 1910 of ten of the longest railroads 
in the United States and, in comparison with them, a correspond- 
ing statement regarding ten more roads selected at random, except 
in the respect that each had a mileage less of than 100 miles. 
Although the extreme variations are greater, yet there is no very 
marked difference in the general values for operating expenses 
per train-mile, or in the ratio of expenses to earnings. The 
averages for the ten long roads agree fairly well with the averages 



380. 



OPERATING EXPENSES. 



457 



for the whole country, but there would be no trouble (as is 
shown by some of the individual cases) in finding another group 
of ten short roads giving either greater or less average values than 
those given. And yet the tendency to uniform values, regard- 
less of the mileage, is very striking. For comparison the figures 
for 1910 have been merely added to the figures for 1904, given 
in previous editions. 



OPERATING EXPENSES PER TRAIN-MILE ON LARGE AND SMALL 
ROADS (1904 AND 1910). 



Mileage. 



1904. 



1910. 



Operating 

expenses per 

train-mile. 



1904. 



1910. 



Ratio expenses 

to earnings 

per cent. 



1904. 1910. 



Whole United States . 



220,112 



240,439 



1.314 



Canadian Pacific 

C, B. & Q 

Chicago & Northwestern . 

Southern Railway 

C, R. I. & P 

Northern Pacific 

A., T. & vS. F 

Great Northern 

Illinois Central 

Atlantic Coast Line 



8,332 
8,326 
7,412 
7,197 
6,761 
5,619 
5,031 
4,489 
4,374 
4,229 



10,271 
9,040 
7,629 
7,050 
7,396 
6,189 
7,460! 
7,147 
4.551 
4,491 



320 
313 



1.136 



.048 
199 



1.302 



Average of ten . 



Montpelier & Wells River. . . 

Somerset Railway Co.* 

Huntingdon & Broadtop 
Mountain 

Lehigh & New England 

Ligonier Valley 

Newburgh, Dutchess & Con- 
necticut t 

Susquehanna & New York . . 

Detroit & Charlevoix 

Harriman & Northeastern * 

Galveston, Houston & Hen- 
derson 



11 

59 
55 
51 
20 

50 



50 
94 

70 

170 
16 



80 
51 
20 

50 



Average of ten (or nine) 1.257 1.539 68.89 74.61 



305 
464 
107 



0.984 



1.227 



1.169 
0.802 

. 950 
0.793 
1.427 

0.922 
1.368 
1.424 
2.162 

1.556 



1.489 



67.79 



66.29 



1.504 

1.710 

1.306 

1.234 

1.344 

1.824 

1.626 

1.^ 

1.409 

1.213 



1.498 



1.430 
1.314 

2.052 
2.04. 
1.480 



1.028 
1.010 
1.733 

1.759 



68.72 

64.35 

66.61 

70.30 

72.90 

52.26 

60.05 

49. 

70.02 

58.95 



63.39 



80.73 
59.37 

52.10 
69 . 80 
69.33 

85.09 
78.47 
67.52 
79.26 

47.27 



65.41 
71.71 
70.31 
67.43 
73.07 
61.71 
64.33 
60.53 
74.84 
62.44 



67.18 



75.08 
76.65 

96.40 
62.84 
49.15 



77.81 
99.53 
63.70 

70.37 



* Subsidiary road since 1904. 

t Merged since 1904; separate figures not available. 



The fluctuations of the average cost per train-mile for several 
years past may be noted from the following tabular form: 



458 



RAILROAD CONSTRUCTION. 



§ 380. 





Average cost 




Average cost 




Average cost 


Year. 


per train-mile 


Year. 


per train-mile 


Year. 


per train-mile 




in cents. 




in cents. 




in cents. 


1890 


96.006 


1897 


92.918 


1904 


131.375 


1891 


95.707 


1898 


95.635 


1905 


132.140 


1892 


96.580 


1899 


98.390 


1906 


137.060 


1893 


97.272 


1900 


107.288 


1907 


146.993 


1894 


93.478 


1901 


112.292 


1908 


147.340 


1895 


91.829 


1902 


117.960 


1909 


143.370 


1896 


93.838 


1903 


126.604 


1910 


148.865 



The enforced economies after the panic of 1893 are well 
shown. The reduction generally took the form of a lowering 
of the standards of maintenance of way and of maintenance of 
equipment. The marked advance since 1895 is partly due to 
the necessity for restoring the roads to proper condition, re- 
plenishing worn-out equipment and providing additional equip- 
ment to handle the greatly increased volume of business. The 
recent advance is chiefly due to the increase in wages and the 
generally increased cost of supplies. 

In looking over the list, it may be noted that the cases where 
the operating expenses per train-mile and the ratio of expenses 
to earnings vary very greatly from the average are almost 
invariably those of the very small roads or of "junction roads" 
where the operating conditions are abnormal. For example, one 
little road, with a total length of 13 miles and total annual opera- 
ting expenses of $5342, spent but 22 Jc. per train-mile, which pre- 
cisely exhausted its earnings. This precise equality of earnings 
and expenses suggests jugglery in the bookkeeping. As another 
abnormal case, a road 44 miles long spent $3.81 per train-mile, 
which was nearly fourteen times its earnings. In another case a 
road 13 miles long earned $7.76 per train-mile and spent $6.03 
(78%) on operating expenses, but the fixed charges were abnor- 
mal and the earnings were less than half the sum of the operating 
expenses and fixed charges. The normal case, even for the 
small road, is that the cost per train-mile and the ratio of operat- 
ing expenses to earnings will agree fairly well with the average, 
and when there is a marked difference it is generally due to 
some abnormal conditions of expenses or of earning capacity. 

381. Reasons for uniformity in expenses per train-mile. 
The chief reason is that, although on the heavy-traffic road 
everything is kept up on a finer scale, better roadbed, heavier 
rails, better rolling stock, more employees, better buildings, 



§ 384. OPERATING EXPENSES. , 459 

stations, and terminals, etc., yet the number of trains is so much 
greater that the divisor is just enough larger to make the average 
cost about constant. This is but a general statement of a fact 
which will be discussed in detail under the different items of 
expense. 

382. Detailed classification of expenses with ratios to the 
total expense. The Interstate Commerce Commission now 
publishes each year a classification with detailed summation 
for the cost of each item. These summations are made up 
from reports furnished by railroads which have (in the reports 
recently made) represented over 99% of the total traffic han- 
dled. In the annexed tabular form (Table XX) are shown the 
percentages which each item bears to the total. The railroads 
have been divided into two classes, '4arge'^ and '^ small," as 
indicated below. Large roads report on 116 items which are 
combined and condensed with 44 items for small roads. 

^' Large roads" are those with mileage greater than 250 miles, 
or those with operating revenues greater than $1,000,000. 
Roads subsidiary to ^' large roads" are also included in this 
class. 

^^ Small roads" are those with mileage less than 250 miles 
and also with operating revenues less than $1,000,000. 

383. Amounts and percentages of the various items. The 
I. C. C. report for the year ending June 30, 1909, was the first 
to include the distribution of expenses according to the present 
classification. The items as given are reliable and may be utihzed, 
as far as any such computations are to be depended on, in 
estimating future expenses. The chief purpose of this dis- 
cussion is to point out those elements of the cost of operating 
trains which may be affected by such changes of location as an 
engineer is able to make. There are some items of expense with 
which the engineer has not the slightest concern, nor will they 
be altered by any change in ahgnment or constructive detail 
which he may make. In the following discussion such items 
will be passed over with a brief discussion of the sub-items 
included. 

MAINTENANCE OF WAY AND STRUCTURES. 

384. Items 2 to 5. Track material. The relative cost of 
ballast, ties, rails and other track material, as shown by com- 
paring either the gross amounts or the percentages in Table IX, 



460 



EAILROAD CONSTRUCTION. 



§384. 



is suggestive and instructive. The fact that ties cost con- 
siderably more than all other track material combined shows 
the importance of any possible saving in tie renewals. It is 
also significant that the relative importance of ties has increased 
in the last few years, and that the relative increase has not been 
due to a reduction in the cost of other track material. Appar- 
ently the lengthening of the average life of ties, due to pre- 
servative processes, the use of tie-plates, and greater care to 
avoid the premature withdrawal- from the track of ties which 
are still serviceable, has not kept pace with the increase in the 
average cost per tie. The cost of Bessemer rails per ton has 
remained almost constant for several years, but the adoption 
of heavier rails by many roads, necessitated by heavier rolling 
stock, and also the adoption of open-hearth steel costing about 
$2 per ton more for much of the tonnage, have been potent causes 
in increasing the cost of maintenance.. 

385. Item 6. Roadway and track. This item is three-eighths 
of the total cost of maintenance of way and structures. It 
consists chiefly of the wages of trackmen. There has been an 
almost steady increase in the daily wages of section foremen 
and other trackmen since 1900, as shown below: 



1900 



1901 



1902 



1903 



1904 



1905 



1906 



1907 



1908 



1909 1910 



Section foremen . . 

Other trackmen . . 

No. of trackmen 

per 100 miles. . . 



1.68 
1.22 



118 



1.71 
1.23 



122 



1.72 
U.25 



140 



1.78 
1.31 



147 



1.78 
1.33 



136 



1.79 
1.32 



143 



1.80 
1.36 



155 



1.90 
1.46 



162 



1.95 
1.45 



130 



1.96 
1.38 



136 



1.99 
1.47 



157 



The average number of section foremen per 100 miles of line 
has remained almost constant at 18. Although there have been 
fluctuations in the number of ^^ other trackmen" required per 
100 miles of line, there has been in general a very substantial 
increase. These two causes combined (increased number and 
increased wages) have had a great influence in producing the 
regular and steady increase in the average cost of a train-mile, 
as shown in § 380. The variations in average daily wages in the 
various Groups for 1910 are shown herewith: 



AVERAGE WAGES 


OF 


TRACKMEN IN 


THE 


SEVERAL GROUPS 


• 




I. 


II. 


m 
III. 


IV. 


V. 


VI. 


VII. 


VIII 


IX. 


X. 


U.S. 


Section foremen . . 
Other trackmen . . 


2.38 
1.65 


2.08 
1.52 


1.99 
1.57 


1.77 
1.21 


1.83 
1.13 


1.90 
1.56 


2.22 
1.60 


1.83 
1.37 


1.88 
1.26 


2.51 
1.52 


1.99 
1.47 



§ 388. OPEEATING EXPENSES. 461 

The classification of expenses for ''small roads" combine 
items 2 to 7. It should be noted that this item (see Table XX A) 
is over three-fourths of the total for that division, and also that 
the cost of maintenance of way and structures for ''small roads" 
is nearly 26% of the total operating expenses and that it is about 
20% for "large roads." In 1908 and 1909, this difference was 
even more marked, and the fact should not be neglected when 
considering the economics of a project which will evidently 
be, at least for many years, a "small road." 

386. Items 8 to 15. Maintenance of track structures. As a 
matter of economics, the locating engineer has Httle or no concern 
with the cost of maintaining track structures. If he is com- 
paring two proposed routes it would be seldom that they would 
be so different that he would be justified in attempting to compute 
a train-mile difference in cost of operation, based on differences 
in these items. Of coiurse, one proposed Hne might call for one 
or more tunnels which the alternate line might not have, and 
the annual cost of maintaining the tunnels would increase the 
cost of operation. Such a case would justify special considera- 
tion. So far as the maintenance of small bridges and culverts 
are concerned it would usually be sufficiently accurate to consider 
that a proposed change of line, involving perhaps several miles 
of road, would require substantially the same number of bridges 
and culverts, and therefore that the cost of maintaining them 
would be the same by either Hne. The error involved in such 
an assumption would usually be insignificant, unless there was 
a very large and material difference in the two lines in this 
respect. Under such conditions special computations should be 
made. The items total less than 3% for smaU roads and still 
less for large roads. 

387. Items 16 to 21. The other items of maintenance of 
way and structures are very small, except No. 16, and under 
ordinary conditions will not be affected by any changes which 
the locating engineer may make, except as noted in the chapter 
on Distance. 

MAINTENANCE OF EQUIPMENT. 

388. Item 24. Superintendence of Equipment. The item 
averages about two-thirds of one per cent and has so Httle 
fluctuation under ordinary conditions that it may almost be 
considered as a fixed charge. It includes those fixed charges in 



462 RAILROAD CONSTRUCTION. § 388. 

superintendence which do not fluctuate with small variations 
in business done. It includes the salaries of superintendent 
of motive power, master mechanic, master car-builder, fore- 
man, etc., but does not include that of road foreman of 
engines nor enginemen. Although the item will vary with 
the general scale of business on the road, it does not fluctu- 
ate with it and hence will not usually be influenced by any 
small changes in alinement which the engineer may be con- 
sidering^. 

389. Items 25 to 27. Repairs, renewals, and depreciation of 
steam locomotives. This subject will be treated at greater 
length in Chapter XXI, Distance. The item is^ of interest 
to the locating engineer because he must appreciate the effect on 
locomotive repairs and renewals of an addition to distance. 
A large part of the repairs of locomotives are due to the wear 
of wheels, which is largely caused by eurvature. Therefore the 
value of any reduction of curvature is a matter of importance, 
and this will be considered in Chapter XXII. A considerable 
portion of the deterioration of a locomotive is due to grade, and 
the economic advantages of reductions of grade will be con- 
sidered in Chapter XXIII. 

This item includes the expenses of work whose effect is sup- 
posed to last for an indefinite period. It does not include the 
expense of cleaning out boilers, packing cylinders, etc., which 
occurs regularly and which is charged to items 72 or 81. It 
does include all current repairs, general overhauling, and even 
the replacement of old and worn-out locomotives by new ones 
to the extent of keeping up the original standard and number. 
Of course additions beyond this should be considered as so 
much increase in the original capital investment. As a loco- 
motive becomes older the annual repair charge becomes a larger 
percentage on the first cost, and it may become as much as one- 
fourth and even one-third of the first cost. When a locomotive 
is in this condition it is usually consigned to the scrap-pile; the 
annual cost for maintenance becomes too large an item for its 
annual mileage. The effect on expenses of increasing the weight of 
engines is too complicated a problem to be solved accurately, but 
certain elements of it may be readily computed.' 'While the cost of 
repairs is greater for the heavier engines, the increase is only 
about one-half as fast as the increase in weight — some of the sub- 
items not being increased at all. 



§ 392. OPERATING EXPENSES. 463 

390. Items 28 to 30. Repairs, renewals, and depreciation of 
electric locomotives. The use of electric locomotives as an ad- 
junct to steam railroad service applies to only a small percentage 
of the roads of the coimtry at present, but the general prin- 
ciples stated in the previous article will apply to these items. 

391. Items 31 to 36. Repairs, renewals, and depreciation of 
passenger-train cars and of freight-cars. Many of the economic 
features of car construction, especially the effect of modern 
improvements, such as friction draft-gear, automatic couplers, 
and air-brakes, have been considered in Chapter XV. Such 
figures will be utiHzed in considering the effect on car repairs 
of additional distance, of variations in curvature, and of grade 
as discussed later. Although the pubhshed figures for these 
items since 1908 are on a shghtly different basis from those 
pubhshed previously, it is evident that these items with respect 
to passenger-cars have remained fairly constant, while those for 
freight-cars have substantially increased, not only in gross 
amount but even in percentage to the total. The fluctuations of 
these items are largely due to accidents rather than to any fine 
of policy under the control of the engineer. It should be under- 
stood that for these items, as well as the previous item, the 
renewal of rolling stock generally means the construction of 
higher-grade locomotives and higher-grade cars. When this is 
done the difference in cost between the higher-grade, and prob- 
ably more expensive, construction and the former cheap con- 
struction should be considered as an addition to the capital 
account rather than a charge against operating expenses. The 
enormous increase in the movement of freight during the last 
few years has required a corresponding increase in freight-car 
equipment. Some roads have been charging up the full cost of 
the renewing of their old-fashioned Hght-weight equipment with 
more expensive equipment imder the head of operating expenses, 
and it is quite possible that a portion of the increase in these 
items is due to this poHcy. 

392. Items 37 to 52. Electric equipment ; floating equipment; 
work equipment; shop machinery and tools; power-plant 
equipment; miscellaneous items. The location of the road 
along the line has no connection with the maintenance of marine 
equipment. The maintenance of shop machinery and tools can 
only be affected as the work of repairs of rolling stock fluctuates, 
and of course in a much smaller ratio. No_change which an 



464 KAILROAD CONSTRUCTION, § 392. 

engineer can effect will have any appreciable influence on this 
item. 

The other items are too small and have too little connection 
with location to be here discussed, except as it may be con- 
sidered that they vary with train-mileage, which an engineer 
may influence (see Chapter XXIII, Grades). 



TKAFFIC. 

393. Items 53 to 60. These items have exclusive reference 
to the work of securing business for the road and have no 
necessary relation to any questions which the locating engineer 
must answer. 

TRANSPORTATION. 

394. Items 61 to 70. These items cover the superintendence 
of transportation and many of the yard and station expenses. 

They require over one-eighth of the entire operating expenses. 
Although some of them might be somewhat affected by the 
details of the design of a yard, they are practically unaffected by 
any changes of alinement which an engineer may make. 

395. Items 71 to 76. Yard-engine expenses. By comparing 
these items with the corresponding items (80 to 85) for road 
engines, it may be seen that the total expenses assignable to 
yard engines are about 20% of those road of engines; the relative 
fuel charge for 1910 was 15.3%. The number of switching 
locomotives in the United States in 1910 was 9115 or 15.4% of the 
total number, 58,947. The relative charge for wages of engine- 
men was 25.7%. This higher proportionate charge is probably 
due to the fact that the wages for yard enginemen must neces- 
sarily be on a per diem basis, but the wages of road enginemen 
are generally on a mileage basis, as explained later. On the 
other hand, the mileage of a yard engine is usually comparatively 
low, and the coal consumed will be correspondingly, although 
not proportionately, low. It must also be remembered that these 
figures are exclusive of the work and equipment of switching and 
terminal companies. I 

396. Item 80. Road enginemen. This item requires 6% of 
the total operating expenses. The enginemen are usually paid 
on a mileage basis, or by the trip, except on very email railroads. 



§ 396. 



OPERATING EXPENSES. 



465 



On very short roads, where a train crew may make two, three, 
or even four complete round trips per day, they may readily be 
paid by the day, so many round trips being considered as a 
day's work, but on roads of great length, where all trains, and 
especially freight-trains, are run day and night, weekday and 
Sunday, all trainmen are necessarily paid by the trip or, as it 
is more usually expressed, by the ^'run.'' It is generally found 
convenient to divide the road into ^^ divisions" which are approx- 
imately 100 miles in length. According as the division is greater 
or less than 100 miles, it is designated as a IJ run, IJ run, or 
perhaps | run. The enginemen will then be paid according to 
the number of runs made per month. There is a considerable 
fluctuation in the average wages paid in different sections of the 
United States, as is shown by the tabular form given below, in 
which the ^^ groups" refer to the sections into which the country 
has been divided by the Interstate Commerce Commission. 
This tabular form may be of some assistance in showing the 
average daily compensation which must be allowed for in the 
various sections of the country: 



AVERAGE DAILY COMPENSATION BY GROUPS — 1910. 





I. 


II. 


III. 


IV. 


V. 


VI. 


VII. 


VIII 


IX. 


X. 


U.S. 


Enginemen 

Firemen 


$ 
3.97 
2.35 


$ 
4.53 
2.75 


$ 
4.34 
2.57 


$ 
4.34 
2.16 


$ 
4.81 
2.42 


$ 
4.58 
2.84 


$ 
4.72 
3.14 


$ 
4.75 
3.13 


$ 
4.87 
3.03 


$ 
4.84 
3.04 


$ 
4.55 
2.74 



The increase in the average wages paid to enginemen and 
firemen during eleven years is plainly shown by the following 
figures: 



INCREASE IN DAILY WAGES, FROM 


1900 


TO 


1910 








1900 


1901 


1902 


1903 


1904 


1905 


1906 


1907 


1908 


1909 


1910 


Enginemen 

Firemen 


$ 
3.75 
2.14 


$ 
3.78 
2.16 


$ 
3.84 
2.20 


$ 
4.01 

2.28 


$ 
4.10 
2.35 


$ 
4.12 
2.38 


$ 
4.12 
2.42 


$ 
4.30 
2.54 


$ 
4.45 
2.64 


$ 

4.4:4: 

2.67 


$ 
4.55 

2.74 



The fluctuations of this item are of importance to the locating 
engineer in discussing the economics of differences in distance* 
This feature will be more fuUy discussed in Chapter XXI. 



466 RAILROAD CONSTRUCTION. § 397. i 

397. Item 82. Fuel for road locomotives. This item in- 
cludes every subitem of the entire cost of the fuel until it is 
placed in the engine-tender. The cost therefore includes not only ] 
the first cost at the point of delivery to the road, but also the 1 
expense of hauling it over the road from the point of delivery [ 
to the various coahng-stations and the cost of operating the j 
coal-pockets from which it is loaded on to the tenders. Even I 
though the cost may be fairly regular for any one road, the cost j 
for different roads is exceedingly variable. There has been an ' 
almost steady increase in the percentage of the cost of this item j 
per train-mile since 1897. Items 73 and 82 amounted to nearly | 
12%|of the total operating expenses in 1910, and required an | 
actual expenditure of nearly $213,000,000. It is the largest | 
item in the whole cost of railroad operation. Although some I 
roads, which traverse coal-regions and perhaps actually own the I 
coal-mines, are able to obtain their coal for a cost which may be I 
charged up as $1 per ton or less, there are many roads which I 
are far removed from coal-fields which have to pay $3 or $4 *| 
per ton, on account of the excessive distance over which the coal ^ 
must be hauled. Unfortunately the figures published by the | 
Interstate Commerce Commission do not show the variations | 
in the percentage of this item in the different groups. In j 
the succeeding chapters will be shown the effect on fuel con- '' 
sumption of the several variations on location details. The ; 
great importance of this item requires that it shall be thoroughly 
understood and studied by the engineer. It will be shown, \ 
contrary to the commonly received opinion, that the fuel con- 
sumption is quite largely independent of distance and even of 
the number of cars hauled. 

398. Items 83 to 95. Water, lubricants, and other supplies 
for road locomotives. These items aggregate about 1.3% of j 
the operating expenses. There seems to be slight tendency for 
the percentage to increase. Since the consumption of all these 
supplies will vary nearly as the engine mileage, the engineer is I 
concerned with them directly to the extent to which he may 
change the engine mileage. "' j 

399. Item 88. Road trainmen. This item includes the wages 1 
of conductors and '^ other trainmen." As in the case of all I 
other employees, the average daily wages have advanced since . 
1900 as shown below: i 



§401. 



OPERATING EXPENSES. 



467 



AVERAGE DAILY WAGES OF CONDUCTORS AND OTHER TRAINMEN, 

, 1900 TO 1910. 





1900 


1901 


1902 


1903 


1904 


1905 


1906 


1907 


1908 


1909 


1910 


Conductors 


$ 
3 17 


$ 
3.17 
2.00 


$ 
3.21 
2.04 


$ 
3.38 
2.17 


$ 
3.50 
2.27 


$ 
3.50 
2.31 


$ 
3.51 
2.35 


$ 
3.69 
2.54 


$ 
3.81 
2.60 


$ 
3.81 
2.59 


$ 
3 91 


Other trainmen. . . 


1.96 


2.69 



The following form shows the variation in the daily wages of 
conductors and trainmen in the several groups for 1910: 



AVERAGE DAILY WAGES IN 1910 IN THE SEVERAL GROUPS. 





I. 


II. 


III. 


IV. 


V. 


VI. 


VII. 


VIII 


IX. 


X. 


U.S. 


Conductors 

Other trainmen.. . 


$ 
3.52 
2.49 


$ 
3.74 
2.71 


$ 
3.73 
2.72 


$ 
3.38 
1.89 


$ 
3.79 
2.18 


$ 
4.12 
2.81 


$ 
4.16 
2.86 


$ 
4.38 
2.84 


$ 
4.59 
3.00 


$ 
4.38 
3.04 


$ 
3.91 
2.69 



These figures are of vital importance from an economic stand- 
point, since they show a constant tendency to increase and 
thereby raise the average cost of a train mile. And as there is 
no present indication of any limit to this increase, aU economic 
calculations which attempt to predict future expenses, even for 
a few years in advance, must allow for these and other increased 
expenses. 

400. Item 89. Train supplies and expenses. These items, 
which average about 1.8%, include the large Hst of consumable 
supphes such as lubricating oil, illuminating-oil or gas, ice, fuel 
for heating, cleaning materials, etc., which are used on the cars 
and not on the locomotives. The consumption of some of these 
articles is chiefly a matter of time. In other cases it is a function 
of mileage. The effect of changes which an engineer may make 
on this item wiU be considered when estimating the effect of the 
changes. 

401. Items 93, 99 to 103. Clearing wrecks, loss, damage and 
injuries to persons and property. These expenses are fortuitous 
and bear no absolute relation either to the number of miles of 
road or the number of train-miles. While they depend largely 
on the standards of discipline on the road, even the best of roads 
have to pay some small proportion of their earnings to these 



468 BAILKOAD CONSTKUCTION. § 401. 

items. While we might expect that a road with heavy traffic 
would have a larger proportion of train accidents than a road of 
light traffic, it is usually true that on the heavy-traffic roads the 
precautions taken are such that they are usually freer from acci- 
dents than the light-traffic roads. Diuring recent years there 
has been a very perceptible increase in the percentages of these 
items, particularly in the compensations paid for ^^ injuries to 
persons.'^ The increase in this item coincides with the increase 
already noted in the number of passengers killed during recent 
years. The possible relation between curvature and accidents 
has already been discussed, but otherwise the locating engineer 
has no concern with these items. 

402. Items 104, 105. Operating joint tracks and facilities, 
Dr. and Cr. A large part of these debit and credit charges 
are those for car per diem and mileage charges. This is a charge 
paid by one road to another for the use of cars, which are chiefly 
freight-cars. To save the rehandling of freight at junctions, 
the policy of running freight-cars from one road to another is 
very extensively adopted. Since the foreign road receives its 
mileage proportion of the freight charge, it justly pays to the road 
owning the car at a rate which is supposed to represent the 
value of the use of the freight-car for the number of miles 
traveled. The foreign road then loads up the freight-car with 
freight consigned to some point on the home road and sends it 
back, paying mileage for the distance traveled on the foreign 
road, a proportional freight charge having been received for that 
service. All of these movements of freight-cars are reported 
to a car association, which, by a clearing-house arrangement, 
settles the debit and credit accounts of the various roads with 
each other. Such is the simple theory. In practice the cars are 
not sent back to the home road at once, but wander off according 
to the local demand. As long as a strict account is kept of the 
movements of every car, and as long as the home road is paid 
the charge which really covers the value of lost service, no harm 
is done to the home road, except that sometimes, when business 
has suddenly increased, the home road cannot get enough cars 
to handle its own business. The value of the car is then abnor- 
mally above its ordinary value, and the home road suffers for 
lack of the roUing stock which belongs to it. Formerly such 
charges were paid strictly according to the mileage. This 
developed the intolerable condition that loaded cars would be 



§ 402. OPERATING EXPENSES. 469 

run onto a siding and left there for several days, simply because 
it was not convenient to the consignee to unload the car imme- 
diately. On the mileage basis the car would be earning nothing, 
and, since the road on which the car then was had no particular 
interest in the car, the car was allowed to stand to suit the con- 
venience of the consignee. To correct this evil a system of per 
diem charges has been developed, so that a railroad has to pay a 
per diem charge for every foreign car on its Hues. To reduce this 
charge as much as possible the railroads compel consignees, 
under penalty of heavy demurrage charges, to unload cars 
promptly. The running of freight-cars on foreign Hues is now 
settled almost exclusively on the per diem basis, but the earning 
of passenger-cars over other hues, as is done on account of the 
advantages of through-car service, as well as the running of 
Pullmans and other special cars, is still paid for on the mileage 
basis. To the extent to which this charge is settled on the mile- 
age basis, any change in distance which the engineer may be able 
to effect in the length of the road will have its influence on this 
item, but when the freight-car business, w^hich comprises by far 
the larger part of the running of cars over foreign Hues, is settled 
on the per diem basis no changes in ahnement which the engineer 
may make will affect the item appreciably. 

Switching Charges. Where two or more railroads intersect 
there will be a considerable amount of shifting of cars, chief!}'' 
freight-cars, from one road to the other. This shifting at any 
one junction may be done entirely by the engines of one road 
or perhaps by those of both roads. A portion of the expense 
of this work is charged up against the other road by the road 
which does the work. The total amount of this work is care- 
fully accounted for by a clearing-house arrangement, and the 
balance is charged up against the road which has done the least 
work. The item is very small, is fairly uniform year by year, 
and is seldom, if ever, affected by changes of alinement. 

Other Items. All of the remaining items, as stated in Table IX, 
are of no concern to the locating engineer. They are either general 
expenses, such as the salaries of general officers, insurance or 
law expenses, or are special items, such as advertising or the 
operation of marine equipment which will not be changed by 
any^variations in distance, curvature, or grades which a locating 
engineer may make. There is therefore no need for their further 
discussion here. 



470 



EAILEOAD CONSTRtrCTION. 



§402. 



TABLE XX. — SUMMARY SHOWING CLASSIFICATION OF OPERATING 
EXPENSES FOR THE YEAR ENDING JUNE 30, 1910, AND PRO- 
PORTION OF EACH CLASS TO THE TOTAL. — ^LARGE ROADS. 



Operating expenses. 



Amount. 



Maintenance of Way and Structures. 

Superintendence of way and structures. . 

Ballast 

Ties 

Rails 

Other track material 

Roadway and track 

Removal of snow, sand, and ice 

Tunnels 

Bridges, trestles, and culverts 

Over and under grade crossings 

Grade crossings, fences, cattle guards, and signs. 

Snow and sand fences and snowsheds 

Signals and interlocking plants 

Telegraph and telephone lines 

Electric power transmission 

Buildings, fixtures, and grounds 

Docks and wharves 

Roadway tools and supplies 

Injuries to persons 

Stationery and printing 

Other expenses 

Maintn'g joint tracks, yds. , and other facilities, Dr 
Maintn'g joint tracks, yds., and other facilities, Cr 

Total maintenance of way and structures . . . . 
Maintenance of Equipment. 

Superintendence of equipment 

Steam locomotives — repairs 

Steam locomotives — renewals 

Steam locomotives — depreciation 

Electric locomotives — repairs 

Electric locomotives — renewals 

Electric locomotives — depreciation » . . . 

Passenger-train cars — repairs 

Passenger-train cars — renewals 

Passenger-train cars — depreciation 

Freight-train cars — repairs 

Freight-train cars — renewals 

Freight-train cars — d epreciation 

Electric equipment of cars — repairs 

Electric equipment of cars — renewals 

Electric equipment of cars — depreciation 

Floating equipment — repairs 

Floating equipment — renewals 

Floating equipment — depreciation 

Work equipment — repairs 

Work equipment — renewals 

Work equipment — depreciation 

Shop machinery and tools 

Power plant equipment 

Injuries to persons 

Stationery and printing 

Other expenses 

Maintaining joint equipment at terminals Dr . . . 
Maintaining joint equipment at terminals, Cr . . . 



117,058,473 

8,861,334 

55,259,585 

16,435,349 

20,225,436 

134,275,440 

8,297,715 

1,147,140 

30,471,313 

1,086,702 

6,105,192 

385,730 

8,175,641 

3,407,627 

367,400 

32,181,561 

3,537,331 

5,141,983 

1,886,865 

714,637 

333,056 

12,151,280 

9,241,467 



^358,265,293 



^11,457,025 

138,548,039 

3,272,370 

11,755,130 

217,182 



22,341 

30,695,229 

1,608,124 

5,827,607 

137,846,373 

12,423,890 

30,712,546 

161,624 

27,000 

42,996 

924,976 

48,144 

381,129 

4,461,503 

754,536 

899,128 

9,439,056 

167,182 

1,370,388 

978,744 

841,019 

1,392,411 

840,895 



Total maintenance of equipment $405,434,797 22.738 



§402. 



OPEEATING EXPENSES. 



471 



TABLE XX. — {Continued.) 



Operating expenses. 



Amount. Percent 



Traffic. 

Superintendence of trafl&c 

Outside agencies 

Advertising 

Traffic associations 

Fast freight lines 

Industrial and immigration bureaus. 

Stationery and printing 

Other expenses 



Total traffic expenses . 



Transportation. 



Superintendence of transportation 

Dispatching trains 

Station employees • . . -, ; • ; 

Weighing and car-service associations 

Coal and ore docks 

Station supplies and expenses 

Yardmasters and their clerks 

Yard conductors and brakemen 

Yard switch and signal tenders 

Yard supplies and expenses 

Yard enginemen 

Enginehouse expenses — yard 

Fuel for yard locomotives 

Water for yard locomotives .' 

Lubricants for yard locomotives 

Other supplies for yard locomotives 

Operating joint yards and terminals — Dr 

Operating joint yards and terminals — Cr 

Motormen 

Road enginemen 

Enginehouse expenses — road 

Fuel for road locomotives 

Water for road locomotives 

Lubricants for road locomotives 

Other supplies for road locomotives 

Operating power plants. . 

Purchased power 

Road trainmen 

Train supplies and expenses _. . . . 

Tnterlockers & block & other signals — operation. 

Crossing flagmen and gatemen 

Drawbridge operation 

Clearing wrecks 

Telegraph and telephone — operation 

Operating floating equipment 

Express service ....._ 

Stationery and printing 

Other expenses 

Loss and damage — freight , . . » 

Loss and damage — baggage . 

Damage to property 

Damage to stock on right of way 

Injuries to persons 

Operating joint tracks and facilities — Dr 

Operating joint track and facilitiess — Cr 



$13,960,333 

19,592,049 

8,347,914 

1,519,215 

3,984,644 

931,212 

6,476,861 

120,052 



$54,932,280 



21,365,630 

16,250,243 

123,062,288 

2,389,673 

2,542,882 

10,302,118 

14,563,707 

48,211,995 

3,888,708 

1,249,844 

27,890,126 

8,106,787 

28,306,698 

1,754,359 

576,693 

644,336 

20,875,654 

12,998,087 

477,404 

108,471,661 

30.580,907 

184,588,622 

11,624,756 

3,573,631 

3,708,286 

726,742 

413,783 

114,122,534 

31,795,761 

8,718,693 

6,319,183 

897,495 

4,488,082 

5,820,358 

2,876,653 

597 

8,113,694 

2,060,571 

21,756,671 

360,244 

4,791,324 

3,650,544 

20,139,285 

4,625,115 

4,357,256 



Total transportation expenses $899.328,994 



* Less than 0.0001 per cent. 



472 



RAILROAD CONSTRUCTION. 
TABLE XX. — {Continued.) 



§402. 



Operating expenses. 



Amount. 



Per cent 



106 
107 
108 
109 
110 
111 
112 
113 
114 
115 

116 



Genekal. 
Salaries and expenses of general officers. . . 
Salaries and expenses of clerks and attendants 

General office supplies and expenses 

Law expenses 

Insurance 

Relief department expenses 

Pensions 

Stationery and printing 

Other expenses 

General administration, joint tracks, yards, 

and terminals — Dr 

General administration, joint tracks, yards, 

and terminals — Or 

Total general expenses 

Recapitulation of expenses: 

I. Maintenance of way and structures. 
II. Maintenance of equipment 

III. Traffic expenses 

IV. Transportation expenses 

V. General expenses 

Total operating expenses 



$9,206,835 

25,167,569 

3,295,407 

10,845,738 

7,551,789 

680,843 

2,007,818 

2,853,808 

3,006,289 

655,875 

192,378 



$65,079,593 



$358,265,293 
405,434,797 

. 54,932,280 

899,328,994 

65,079,593 



0.516 
1.411 
0.185 
0.608 
0.424 
0.038 
0.113 
0.160 
0.169 

0.037 

0.011 



3.650 



20.093 
22.738 

3.081 
50.438 

3.650 



11,783,040,957 



100.000 



The "Credit items" 23, 52, 78, 105, and 116 represent receipts from other 
roads, from switching and terminal companies (which are not included in this 
or the following table) or from other sources. The amounts and percentages 
must be deducted from the other expenses to obtain the net expenses. 



TABLE XXa. — SUMMARY SHOWING CLASSIFICATION OF OPERATING 
EXPENSES FOR THE YEAR ENDING JUNE 30, 1910, AND PRO- 
PORTION OF EACH CLASS TO THE TOTAL. SMALL ROADS. 



Operating expenses. 



Per cent 



Correspond- 
ing items in 
Table XX. 



I. Maintenance of Way and Structures 

Superintendence of way and structures 

Maintenance of roadway and track 

Maintenance of track structures 

Maintenance of buildings, docks, and wharves 

Injuries to persons 

Other maintenance of way & structures expenses 
Maintaining joint tracks, yards, and other fa- 
cilities — Dr 

Maintaining joint tracks, yards, and other fa- 
cilities — Cr 

Total, maintenance of way and structures. . 



1.195 
19.752 
2.991 
1.191 
0.042 
0.860 

0.469 

0.668 



1 

2-7 

8-15 

16, 17 

19 

18, 20, 21 

22 

23 



25.832 



§402. 



OPERATING EXPENSES. 



473 



TABLE XXa. — SUMMARY SHOWING CLASSIFICATION OF OPERATING 
EXPENSES FOR THE YEAR ENDING JUNE 30, 1910, AND PRO- 
PORTION OF EACH CLASS TO THE TOTAL. — SMALL ROADS. — 

{Continued.) 



Operating expenses. 



Per cent. 



Correspond- 
ing items in 
Table XX. 



II. Maintenance of Equipment. 

Superintendence of equipment 

Locomotives — repairs 

Cars — repairs 

Floating equipment — repairs 

Work equipment — repairs 

Equipment — renewals 



Equipment — depreciation . 



Injuries to persons 

Other maintenance of equipment expenses . . . . 

Maintaining joint equipment at terminals — Dr . 

Maintaining joint equipment at terminals — Cr. 

Total, maintenance of equipment 



III. Traffic Expenses. 



Traffic expenses. 



IV. Transportation Expenses. 

Superintendence and dispatching trains 

Station service 

Yard enginemen r 

Other yard employees 

Fuel for yard locomotives 

All other yard expenses 

Operating joint yards and terminals — Dr. . . . 
Operating joint yards and terminals — Cr. . . . 

Road enginemen and motormen 

Fuel for road locomotives 

Other road locomotive supplies and expenses. 

Road trainmen 

Train supplies and expenses 

Injuries to persons 

Loss and damage 

Other casualties 

All other transportation expenses 

Operating joint tracks and facilities — Dr.. . . 
Operating joint tracks and facilities — Cr. . . . 



V. General Expenses. 

Administration 

Insurance 

Other general expenses 

General administration, joint tracks, yards, and 

terminals — Dr 

General administration, joint tracks, yards, and 

terminals — Cr 



0.919 
6.275 
6.510 
0.044 
0.165 
0.608 

4.214 

0.025 
0.785 
0.026 
0.063 



19 . 508 



2.418 



2.200 
6.185 
0.823 
1.259 
1.276 
0.376 
0.743 



5.977 
11.184 
2.850 
6.313 
0.936 
0.619 
0.471 
0.616 
1.916 
0.481 
0.135 



43 . 472 



7.088 
0.749 
0.968 

0.031 

0.066 

8.770 



24^ 

25,28 

31, 34, 37 

40 

43 

26, 29, 32, 35, 

38, 41, 44 

27, 30, 33. 36, 

39, 42, 45 
48 

46, 47, 49, 50 
51 
52 



53-60 



61, 62 

63-66 

71 

67-69 

73 

70, 72, 74-76 

77 

78 

79, 80 

82 

81, 83-87 



103 
99, 100 
93, 101, 102 
90-92, 94-981 

104 

105 



106-109 

110 
111-114 



115 
116 



CHAPTER XXL 
DISTANCE. 

403. Relation of distance to rates and expenses. Rates 
are usually based on distance traveled, on the apparent 
hypotheses that each additional mile of distance adds its pro- 
portional amount not only to the service rendered but also to 
the expense of rendering it. Neither hypothesis is true. The 
value of the service of transporting a passenger or a ton of 
freight from A to 5 is a more or less uncertain gross amount 
depending on the necessities of the case and independent of 
the exact distance. Except for that very small part of passen- 
ger traffic which is undertaken for the mere pleasure of traveling, 
the general object to be attained in either passenger or freight 
.traffic is the transportation from A to B^ however it is attained. 
A mile greater distance does not improve the service rendered; 
in fact, it consumes valuable time of the passengers and perhaps 
deteriorates the freight. From the standpoint of service ren- 
dered, the railroad which adopts a more costly construction and 
thereby saves a mile or more in the route between two places 
is thereby fairly entitled to additional compensation rather 
than have it cut down as it would be by a strict mileage rate. 
The actual value of the service rendered may therefore vary 
from an insignificant amount which is less than any reasonable 
charge (which therefore discourages such traffic) and its value 
in cases of necessity — a value which can hardly be measured in 
money. If the passenger charge between New York and Phila- 
delphia were raised to $5, $10, or even $20, there would still be 
some passengers who would pay it and go, because to them 
it would be worth $5, $10, or $20, or even more. Therefore, 
when they pay $2.25 they are not paying what the service is 
worth to them. The service rendered cannot therefore be 
made a measure of the charge, nor is the service rendered pro- 
portional to the miles of distance. 

The idea that the cost of transportation is proportional to 

474 , 



§ 405. DISTANCE. 475 

tlie distance is much more prevalent and is in some respects 
more justifiable, but it is still far from true. This is especially 
true of passenger service. The extra cost of transporting a single 
passenger is but little more than the cost of printing his ticket. 
Once aboard the train, it makes but little difference to the rail- 
road whether he travels one mile or a hundred. Of course there 
are certain very large expenses due to the passenger traffic 
which must be paid for by a tariff which is rightfully demanded, 
but such expenses have but little relation to the cost of an 
additional mile or so of distance inserted between stations. 
The same is true to a slightly less degree of the freight traffic. 
As showTL later, the items of expense in the total cost of a train- 
mile, which are directly affected by a small increase in distance, 
are but a small proportion of the total cost. 

404. The conditions other than distance that affect the cost; 
reasons why rates are usually based on distance. Curvature 
and minor grades have a considerable influence on the cost of 
transportation, as mil be shown in detail in succeeding chap- 
ters, but they are never considered in making rates. Ruling 
grades have a very large influence on the cost, but they are like- 
wise disregarded in making rates. An accurate measure of 
the effect of these elements is difficult and comphcated and 
would not be appreciated by the general public. Mere dis- 
tance is easily calculated; the pubhc is satisfied with such 
a method of calculation; and the railroads therefore adopt a 
tariff which pays expenses and profits even though the charges 
are not in accordance mth the expenses or the ser\dce rendered. 

An addition to the length of the Hne may (and generally does) 
involve curvature and grade as well as added distance. In 
this chapter is considered merely the effect of the added dis- 
tance. The effect of grade and curvature must be considered 
separately, according to the methods outlined in succeeding 
chapters. The additional length considered is likemse assumed 
not to affect the business done nor the number of stations, but 
that it is a mere addition to length of track. 

EFFECT OF DISTANCE ON OPERATING EXPENSES. 

405. Effect of changes in distance on maintenance of way. 

The items of maintenance of way are more nearly affected in 
proportion to the distance than any other group of items. In 



476 RAILROAD CONSTRUCTION. § 405. 

fact it will be easier to note the exceptions from a full 100% 
addition for all increases of distance. The cost of track labor, 
which is such a large percentage of the total cost of Item 6 (see 
Table XX in Chapter XX), and also the cost of all track material, 
will vary almost exactly in accordance with the distance. If 
the track-labor was so perfectly organized that there were no 
more laborers than could- precisely accomplish the necessary 
work, working full time, then any additional labor would nec- 
essarily require a greater expenditure for laborers. Although 
a division of a road is divided into sections of such a length that 
a gang of say six or seven men will be employed as steadily as 
possible in maintaining the track in proper condition, the addi- 
tion of a few feet of track would not probably have the effect 
of increasing the number of sections, nor would it even require 
the addition of another man to the track-gang. It might 
require a little harder work in maintaining a section, it might 
even mean a slight lowering in the standard of work done in 
order that the whole section should be covered. The fact 
remains that the cost of track-labor will not inevitably and 
necessarily be increased in a strict proportion to the increase 
in distance. On the other hand, it would not be wise to rely 
on any definite reduction or discount from the full 100% of 
work required, since to do so implies that, with the lessened 
distance there would be some loafing on the part of the track- 
gang, or that with the added distance the men would be over- 
worked or would be compelled to sHght their work. The items, 
renewals of rails and renewals of ties, should certainly be con- 
sidered as changing in direct proportion to the distance. There- 
fore it is only safe to allow the full 100% addition for Items 
1 to7. 

The repairs and renewals of tunnels, bridges, culverts, fences, 
road crossings, signs, cattle-guards, buildings and fixtures, 
docks and wharves, etc. (Items 8 to 17), may perhaps be con- 
sidered in the same way. although there are some of these items 
on which the effect is more doubtful. If a proposed change in 
line does not involve any difference in the number of streams 
crossed, then the number of the bridges and culverts will not 
be altered, and although the size may be altered, the effect of 
the change on the cost for repairs will probably be too insignif- 
icant for notice. For small changes of distance it may very 
readily happen that no bridge or culvert is involved. For 



§ 406. DISTANCE. 477 

great changes of distance, especially those which would involve 
an entire change of route for a distance of many miles, it might 
be proper to consider Item 9 to be affected fully 100%. Although 
Items 9 and 10 are small, averaging about 1.8%, the error 
involved in these items by considering that the change amounts 
to 100% for great distances and zero for smaU distances will be 
almost inappreciable. For Items 11 to 13 the full 100% will 
be allowed for all changes of distance, for the same reason as 
previously given for repairs of roadway. Item 16 will usually 
be absolutely unaffected by a small change in distance, since 
it does not usually involve any buildings or fixtures. Larger 
changes of distance will probably require some change in the 
number of minor buildings required, but such buildings will be 
the more insignificant buildings, and we are therefore making 
ample allowance, if, under ordinary conditions, we estimate 
that 20% of the average cost of aU buildings (which include 
terminals, etc.) is allowed for this item. Under ordinary con- 
ditions Item 17 will be absolutely unaffected by any changes 
in aHnement which the engineer may make. An addition to 
distance will not usually affect the telegraph system, except 
as it adds to the number of telegraph-poles and to the amount 
of wiring and pole fixtures. Therefore any addition to dis- 
tance will not add more than 50% to the average cost of Item 
14. Items 18 to 21 are insignificant in amount, and can hardly 
be said to be affected by any small difference in distance which 
would ordinarily be measured in feet. Larger differences, 
which are measured in miles and which may involve, for instance, 
all the blank forms required for the reports of an additional 
section-gang, additional pay-rolls, etc., will be increased to 
practically their fuU proportion. Therefore there is but httle 
error involved in allowing 100% on these items for changes of 
distance measured in miles. 

406. Effect on maintenance of equipment. The relation 
between an increase in length of fine and the expenses of Items 
24, and 46 to 50 are quite indefinite. In some respects they 
would be unaffected by slight changes of distance, and yet it 
is difficult to prove that the expenses should not be considered 
proportionate for the distance. For example, the added train- 
mileage will increase repairs of rolHng-stock, and will therefore 
hasten the deterioration and increase the cost of '' repairs and 
renewals of shop machinery and tools '' (Item 46). Fortunately, 



478 RAILROAD CONSTRUCTION. § 406. 

all these items are so small, even in the aggregate, that little 
error will be involved, whatever decision is made. It will 
therefore be assumed that these items are affected 100% for 
large additions in distance and 50% for small additions. Items 
40 to 42 are evidently unaffected by any change of distance. 
Electrical equipment, items 28-30 and 37-39, which are used 
on so few steam railroads, is ignored in this discussion. 

There only remain the four groups of items, the repairs, 
renewals and depreciation of steam locomotives and of passenger 
and freight-cars and of work equipment. The deterioration 
of rolling stock, which requires its repair and finally its ultimate 
abandonment and therefore renewal, is caused by a combina- 
tion of a large number of causes, of which the mere distance 
they travel on the road is but one cause. They deteriorate 
first with age; second, on account of the strains due to stopping 
and starting; third, on account of the strains and wear of 
wheels due to curved track; fourth, on account of the addi- 
tional stresses due to grade and change of grade, and fifth, on 
account of the work of pulling on a straight level track. In 
addition to this, locomotives suffer considerable deterioration 
due to expansion and contraction, especially of the fire-box 
when the fires are drawn and the fire-box and boiler become cold, 
and again when the fire is started up. A large part of the ex- 
penses of maintaining passenger-cars is the expense of painting, 
which is a matter of mere time. Considering that the changes 
of distance, whose economic value the engineer tries to com- 
pute, will never make a difference in the number of round trips 
the engine or car would make in a day or month, the added 
distance which may be traveled does not add to the exposure 
of the car to the weather. Therefore, whatever deterioration 
of the car paint is due to weather, it will be incurred regardless 
of whether the length of the division of the road is 100 miles 
or 99 or 101. That element of the cost of car maintenance is 
absolutely independent of the precise length of that division 
of the road. On the other hand, the wear of car- and engine- 
wheels, although largely affected by curvature, is certainly 
affected to some extent by wear on a straight tangent. To 
determine the proportion of total wear due to these various 
causes is a matter of estimation and judgment. An approach 
to accuracy may be made by a compilation of the shop records 
of rolling-stock, repairs, showing the amour t which is spent 



§406. 



DISTANCE. 



479 



in various kinds of repairs, and estimating as closely as possible 
what is the cause of each form of deterioration. A check on 
any such estimate is the consideration that the total deteriora- 
tion is simply the summation of the deterioration due to all 
causes combined. It is therefore a question of dividing 100% 
into as many portions as there are contributing causes, and to 
assign to each cause its relative importance in per cent, so 
that the sum total shall reach 100. A. M. Wellington, in his 
" Economic Theory of Railway Location,^' distributed the 
cost of engine repairs to its various contributing causes, as 
shown in the following tabular form. He did not claim that 
such an estimate was accurate and applicable to all cases, but 
he did claim that the error was probably not sufficient to be 
of importance. A comparison of these percentages, with the 
data given by shop records on any particular road, would not 
probably show a very material difference, and the writer will 
not attempt to claim that any figures he might obtain will be 
any more accurate in general, although they might be more 
accurate as appHed to some particular conditions. 



DISTRIBUTION OF THE COST OF ENGINE REPAIRS TO ITS VARIOUS 

CONTRIBUTING CAUSES^ (Copied from Wellington.) 





Total 
cost of 
item. 


Distribution. 




Item. 


Effect 
of time, 
age, and 

expos- 
ure, per 

cent. 


Stop- 
ping 
and 
starting 
at way 
stations 
per 
cent. 


Termi- 
nal: 
getting 
up 
steam 
making 

up 
trams, 

per 
cent. 


Curva- 
ture and 
grades, 
per 
cent. 
(App'x- 
imate 
aver- 
age.) 


Dis- 
tance on 
tangent 
between 
sta- 
tions, 
per 
cent. 


Boiler 


20.0 
20.0 
30.0 


"i*" 


2 

4 

7 


7 
2 
3 


4 
7 
5 


7 


Running gear 


7 


Machinery 


14 


Mountings 




Lagging and painting . . 

Smoke-box, etc 

Tender: 
Running gear 


12.0 
5.0 

10.0 
3.0 


4 
1 

*"i*" 


2 


2 
1 

1 

1 


3 


6 
3 

4 


Body and tank 


1 


Total 


100.0 


7 


15 


17 


19 


42 







It may be noted from the above table that 42% of engine 
repairs has been assigned to distance on a tangent between sta- 



480 RAILROAD CONSTRUCTION. § 406. 

tions. If the added distance does not imply an extra stoppage 
of the train, there is but Httle, if any, reason to differentiate 
between the effect on repairs of a large or small addition to 
distance. We will therefore consider Items 25-27 to be affected 
in this ratio of 42%. 

Wellington similarly distributed the cost of freight-car repairs 
to its various causes, and by a very similar method estimated 
that 36% of such repairs was due to distance on a tangent 
between stations. The considerable transformation in the 
construction of freight-cars, since the time that Wellington 
compiled this table, has certainly utterly changed the absolute 
cost of car repairs, even if it has not changed the relative per- 
centage of the cost of the various items. In the lack of any 
better figures this same figure will be for Items 34-36. Although 
there are evidently enormous differences between Items 34-36 
and 43-45, Items 43-45 are so small that it is hardly worth the 
calculation of any more precise figures, and therefore the same 
ratio, 36%, will be used for Items 43-45. Wellington made 
no definite calculations for the itemized cost of passenger-car 
repairs, but contented himself with using the same figure as for 
freight-cars, 36%. Such a percentage is probably very much 
too high, since it is estimated that about one-half the cost 
of passenger-car repairs is due to the work of painting, inside 
and out, and of maintaining the seats and upholstery in proper 
condition. Such repairs are chiefly a function of time, and 
are but Httle, if any, dependent on mere distance between 
stations. It is therefore considered that Items 31-33 will not 
be affected more than 20% by any addition of distance. 

Traffic expenses will be unaffected. 

407. Effect on conducting transportation. Items 61-78, 
superintendence and dispatching, all station and yard expenses, 
and the joint expenses of yards and terminals, may be 
considered as unaffected. Item 79, motormen's wages, is 
ignored. 

Item 80. Road enginemen. In a previous chapter, §396 
the wages of road enginemen were discussed. The discussion 
showed that the enginemen are rarely, although sometimes, 
paid on a strict mileage basis. They are usually paid on a trip 
basis, in accordance with which a slight change in the distance 
will not affect the classification of the trip, and therefore would 
make no difference in the wages. We will therefore say that 



§ 407. DISTANCE. 481 

for small changes of distance, especially such as would be 
measured in feet, this item will be unaffected; but that for 
larger changes, such as would be measured in miles, this item 
will be affected by the full amount of the enginemen's wages. 

Although there might be some justification for saying that 
the engine-house expenses for road engines (Item 81) might 
be somewhat increased by additions to distance, the addition 
may be considered as already covered by the increase in main- 
tenance charges (Items 25-27), and no further allowance will be 
made. 

Item 82. Fuel. A surprisingly large percentage of the fuel 
consumed is not utihzed in drawing a train along the road. 
A portion of this percentage is used in firing-up. A portion is 
wasted when the engine is standing still, which is a considerable 
proportion of the whole time. The poHcy of banking fires 
instead of drawing them reduces the injm-y resulting from great 
fluctuations in temperature, but in a general way we may say 
that there is but Httle, if any, saving in fuel by banking the 
fires, and therefore we may consider that almost a fire-box fuU 
of coal is wasted whether the fires are banked or drawn. Some 
tests were made on the Santa Fe, in which some large locomotives 
consumed from 1200 to 1660 pounds of coal merely in firing-up. 
But even the amount of coal required to produce the required 
steam-pressure in the boiler from cold water does not represent 
the total loss. The train-dispatcher, in his anxiety that engines 
shall be ready when needed, will sometimes order out the loco- 
motives which remain somewhere in the yard, perhaps exposed 
to cold weather, and blow off steam for several hours before 
they make an actual start. Of course the amount thus attribut- 
able to firing-up is a very variable one, depending on the manage- 
ment, and therefore no precise figures are obtainable. But 
it has been estimated that it amounts to from 5 to 10% of the 
total consumption. A freight-train, especially on a single- 
track road, will usually spend several hom*s during the day on 
sidings, and when a single-track road is being run to the limit 
of its capacity, or when the management is not good, the time 
will be still greater. It has been found that the amount of coal 
required by an engine merely to keep up steam will amount to 
from 25 to 50 pounds per hour. If, in addition to this, steam 
escapes through the safety-valve, the loss is much larger. It 
is estimated that the amount lost through a 2|-inch safety- 



482 RAILKOAD CONSTRUCTION. § 407. 

valve in one minute would represent the consumption of 15 
pounds of coal, which would be sufficient to haul 100 tons on a 
mile of track with easy grades. Again we see that the amount 
thus lost is exceedingly variable and almost non-computable, 
although as a rough estimate the amount has been placed 
at from 3 to 6% of the total. Another very large subitem 
of loss of useful energy is that occasioned by stopping and 
starting. A train running 30 miles per hour has enough kinetic 
energy to move it on a straight level track for more than two 
miles. Therefore, every time a train running at 30 miles per 
hour is stopped, enough energy is consumed by the brakes to 
run it about two miles. There is a double loss^ not only due 
to the fact of the loss of energy, but also because the power of 
the locomotive has been consumed in operating the brakes. 
When the train is again started, this kinetic energy must be 
restored to the train in addition to the ordinary resistances which 
are even greater, on account of the greater resistance at very 
low velocities. Of course, the proportion of fuel thus con- 
sumed depends on the frequency of the stops. It was demon- 
strated by some tests on the Manhattan Elevated Road in New 
York City, where the stops average one in every three-eighths 
of a mile, that this cause alone would account for the consump- 
tion of nearly three-fourths of the fuel. On ordinary railroads 
the proportion, of course, will not be nearly so great, but there 
is reason to believe that 10 to 20% is not excessive as an average 
figiu-e. The amount of fuel which is consumed on account of 
curvature is, of course, a function of the curvature and will 
vary with each case. The only possible basis for a calculation 
of this amount must be somewhat as follows: Since all of the 
above calculations consider the average cost of a train-mile 
throughout the country, we must consider what is the average 
amount of curvature per mile of track throughout the country. 
By estimating, as will be developed in the next chapter, the 
proportion of the fuel expenditure which is due to this average 
amount of curvature and subtracting this, as well as the other 
subitems enumerated above, from the total average cost of a 
train-mile, we would then have the desired quantity, the cost 
of fuel per mile of straight level track. Although it is not easy 
to obtain reliable statistics showing the average curvature per 
mile or road throughout the United States, there is reason to 
believe that it is not far from 35° per mile. According to tho 



§ 407. DISTANCE. 483 

method of calculations given in the next chapter, to determine 
the additional fuel consumed by the added resistance due to 
35° of curvature per mile, we obtain the approximate value 
that about 4% of the fuel consumption will be due to this cause. 
To obtain an average figure for the resistance due to grade is 
perhaps even harder, but on the approximate basis that the 
average amount of rise and fall per mile of track is about 11 feet 
per mile, it would seem as if 25% of the consumption of fuel 
were due to grades. Summarizing the above items we would 
have 

Firing 5 to 10% 

Loss by radiation, etc 3 to 6% 

Stopping and starting 10 to 20% 

Average curvature 4 to 4% 

Average grade 25 to 25% 

47 to 65% 

Direct hauling 53 to 35% Average 44% 

-^ 

100 . 100 

This gives, as an average figure for the increased consumption 
of fuel due to one additional mile of straight level track, 44% 
of the average consumption per mile. 

Items 83, 84, and 85. Minor engine-supplies. If water is 
obtained from municipal supphes and paid for at meter rates, 
then the cost will be strictly according to the consumption, 
which will be nearly according to the number of engine-miles. 
Almost the only waste would be that occasioned by blowing off 
steam. Under such circumstances the increase in this item 
would be very nearly 100%, but, on the other hand, where the 
supply is obtained from the company's own plant, there is hardly 
any appreciable increase in expense due to the extra draft on 
the tanks. Of course the cost of pumping would be somewhat 
affected. Considering that the sum of Items 83, 84, and 85 
is only 1.06%, very little error would be involved if we con- 
sider as an average figure that this item will be increased the 
same as fuel, 44%. 

Item 88 includes the wages of trainmen other than engine- 
men. Their wages are paid on very much the same basis as 
the enginemen, which means usually that small additions of 
distance will not affect their wages. Large additions wiU 
affect them 100%; 



484 EAILROAD CONSTRUCTION. § 407. 

Item 89. Train-supplies and expenses include a very large 
matter of small subitems, the consumption of which is partly 
a matter of mere time and partly a matter of mileage. It is 
sufficiently precise in this case to say that 50% of these sub- 
items will be affected directly as the mileage. 

Items 90, 91. Signals, flagmen, and gatemen. The necessity 
for flagmen and switchmen may be said in general to increase 
with the mileage, although it might readily happen that a given 
change in distance which is under consideration might not effect 
the slightest change in this item. It is quite unlikely that the 
number of switchmen would be affected. It is probable that 
25% of this item is sufficient as an average figure. 

Item 94. Telegraph expenses include the wages of operators 
at stations (which are unaffected) and the special expenses 
due to offices and telegraph stations and to operating the line, 
the maintenance of the line being charged to Item 14. Although 
it will theoretically require more battery material to transmit 
telegrams over a longer line, the added expense is so very sHght 
that it may be utterly ignored. 

Items 86 and 87, operating power plants and purchased power, 
which depend on electric traction, are ignored. Item 92, 
drawbridge operation, and Item 95, operating floating equip- 
ment, are considered to be unaffected. Items 96 to 98 are 
small miscellaneous expenses which are considered unaffected, 
as are likewise Items 104 and 105, the operation of joint tracks 
and facilities. This leaves Items 93 and 99 to 103, which are 
the fortuitous expenses due to wrecks, damage and mishaps. 
While it cannot be definitely predicted that these will happen 
with any stated regularity, it is only proper to consider that 
they will increase with added distance and to charge them 
as strictly proportional to distance. 

The general expenses, Items 106 to 116, are considered to 
be unaffected. 

408. Estimate of total effect on expenses of small changes in 
distance measured in feet ; also estimate for distances measured 
in miles. Collecting the above percentages for the various 
items we have Table XXI, which shows that the average cost 
of operating a small additional distance will be about 33% 
of the average cost per unit distance. If the additional dis- 
tance amounts to several miles, the added cost will amount 
to about 50% of the unit cost. These figures may also be conr 



408. 



DISTANCE. 



485 



TABLE XXI. 



EFFECT ON OPERATING EXPENSES OF GREAT AND 
SMALL CHANGES IN DISTANCE. 







Per cent affected. 


Cost per mi 


ie, per cent. 


No. of item. 


Normal 
















average. 














Great. 


Small. 


Great. 


Small. . 


*l-7 


14.61 


100 


100 


14.61 


14.61 


8 


0.06 














9, 10 


1.77 


100 





1.77 





11-13 


0.82 


100 


100 


0.82 


0.82 


14 


0.49 


50 


50 


0.10 


0.10 


15 


0.02 














16 


1.81 


20 





0.36 





17 


0.20 














18-21 


0.45 


100 





0.45 





22, 23 


0.16 
















20.09 






18.11 


15.53 










24 


0.64 


100 


50 


0.64 


0.32 


25-27 


8.62 


42 


42 


3.62 


3.62 


28-30 


0.01 














31-33 


2.14 


20 


20 


0.43 


0.43 


34-36 


10.15 


36 


36 


3.65 


3.65 


37-39 


0.01 














40-42 


0.08 














43-45 


0.34 


36 


36 


0.12 


0.12 


46-50 


0.72 


100 


50 


0.72 


0.36 


51, 52 


0.03 
















22.74 






9.18 


8.50 










53-60 


3.08 














61-79 


17.92 














80 


6.08 


100 





6.08 





81 


1.72 














82-85 


11.41 


44 


44 


5.02 


5.02 


86, 87 


0.06 














88 


6.40 


100 





6.40 





89 


1.78 


50 


50 


0.89 


0.89 


90, 91 


0.85 


25 


25 


0.21 


0.21 


92 


0.05 














93 


0.25 


100 


100 


0.25 


0.25 


94-98 


1.06 














99-103 


2.84 


100 


100 


2.84 


2.84 


. 104, 105 


0.02 
















50.44 






21.69 


9.21 










106-116 


3.65 
















100.00 






48.98 


3 3.24 











* For the significance of the items, see Table XX. 

sidered as the saving in operating expenses resulting from a 
shortening of the hne, and thus gives a measure of the operating 
value of reducing the length of the hne. The average cost of a 
train-mile since 1890 has varied between 91.829 c. in 1895 to 



486 KAILROAD CONSTRUCTION. § 408. 

148.865 c. in 1910. The cost has been rising almost steadily 
since 1897. Whether the cost will continue to rise or whether 
it will recede during the next few years is of course a matter of 
pure conjecture. Even if the cost recedes somewhat from the 
high value of recent years, it is quite certain that it will never 
again sink to the low value of 1895. If we adopt the round 
number of $1.50 as the probable cost of a train-mile during 
the next few years, we can reduce the above percentages to cents 
per train-mile, which will come to 50 and 7^ c. per train-mile 
respectively. Some trains run 365 days per year; others run 
but 313 days. The tendency, however, is toward the larger 
figure, especially in the case of freight service, which comprises 
about 52% of the number of train-miles. The added cost 
per daily train per year for each foot of distance would therefore 
be 

50X365X2 



5280 



=6.91 c. 



When the distance is measured in miles the added cost per 
daily train per year for each mile of distance would be 

.73X365X2 =$533. 

Of course, if such calculations are made for a light traffic road 
which only runs trains on week-days, we should use 313 in the 
above equations instead of 365. It should be noted that the 
subitems in the above table which are the most uncertain are 
those whose absolute value is the smallest, and that even if 
we make very large variations in the most uncertain items, 
the final result will not be very materially altered. On the 
other hand, the very largest items are those which are capable 
of fairly precise calculations. 

EFFECT OF DISTANCE ON RECEIPTS. 

409. Classification of traffic. There are various methods 
of classifying traffic, according to the use it is intended to 
make of the classification. The method here adopted will have 
reference to its competitive or non-competitive character and 
also to the method of division of the receipts on through traffic. 
Traffic may be classified first as ''through'' and ''locaP' — 



§ 411. DISTANCE. 487 

through traffic being that traveling over two (or fmore) lines, 
no matter how short or non-competitive it may be; '^ocal" 
traffic is that confined entirely to one road. A fivefold classifica- 
tion is however necessary — ^w^hich is : 

A. Non-competitive local — on one road with no choice of 
routes. 

B. Non-competitive through — on two (or more) roads, but 
with no choice. 

C. Competitive local — a choice of two (or more) routes, but 
the entire haul may be made on the home road. 

D. Competitive through — direct competition between two 
or more routes each passing over two or more lines. 

E. Semi-competitive through — a non-competitive haul on the 
home road and a competitive haul on foreign roads. 

There are other possible combinations, but they all reduce to 
one of the above forms so far as their essential effect is concerned. 

410. Method of division of through rates between the 
roads run over. Through rates are divided between the 
roads run over in proportion to the mileage. There may 
be terminal charges and possibly other more or less arbitrary 
deductions to be taken from the total amount received, but 
when the final division is made the remainder is divided accord- 
ing to the mileage. On account of this method of division and 
also because non-competitive rates are always fixed according 
to the distance, there results the unusual feature that, unlike 
curvature and grade, there is a compensating advantage in 
increased distance, which applies to all the above kinds of 
traffic except one (competitive local), and that the compensation 
is sometimes sufficient to make the added distance an actual 
source of profit. It has just been proved that the cost of hauling 
a train an additional mile is only 33 to 49% of the average 
cost. Therefore in all non-competitive business (local or 
through) where the rate is according to the distance, there is 
an actual profit in all such added distance. In competitive 
local business, in which the rate is fixed by competition and 
has practically no relation to distance, any additional distance 
is dead loss. In competitive through business the profit or 
loss depends on the distances involved. This may best be 
demonstrated by examples. 

411. Effect of a change in the length of the home road on 
its receipts from through competitive traffic. Suppose the 



488 KAILROAD CONSTRUCTION. § 411. 

home road is 100 miles long and the foreign road is 150 miles 

long. Then the home road will receive -— r- — -^-r =40% of the 

iUU ~r -i-oU 

through rate. 

Suppose the home road is lengthened 5 miles; then it will 

105 
receive ^^^ , ,,^- =41.176% of the through rate. The traffic 
iUo + ioU 

being competitive, the rate will be a fixed quantity regardless 
of this change of distance. By the first plan the rate received 
is 0.4% per mile; adding 5 miles, the rate for the original 100 
miles may be considered the same as before; and that the addi- 
tional 5 miles receive 1 . 176%, or . 235% per mile. This is 59% 
of the original rate per mile, and since this is more than the 
cost per mile for the additional distance (see § 408), the added 
distance is evidently in this case a source of distinct profit. 
On the other hand, if the line is shortened 5 miles, it may be 
similarly shown that not only are the receipts lessened, but 
that the saving in operating expenses by the shorter distance 
is less than the reduction in receipts. 

A second example will be considered to illustrate another 
phase. Suppose the home road is 200 miles long and the foreign 
road is 50 miles long. In this case the home road will receive 

„ =80% of the through rate. Suppose the home road is 

205 
lengthened 5 miles; then it will receive ^r^rp—-rTr = 80.392% 

of the through rate. By the first plan the rate received is 
. 400% per mile ; adding 5 miles, there is a surplus of . 392, 
or 0.0784 per mile, which is but 19.6% of the original rate. 
At this rate the extra distance evidently is not profitable, al- 
though it is not a dead loss — there is some compensation. 

412. The most advantageous conditions for roads forming 
part of a through competitive route. From the above it may 
be seen that when a road is but a short link in a long com- 
petitive through route, an addition to its length will increase 
its receipts and increase them more' than the addition to the 
operating expenses. 

As the proportionate length of the home road increases the 
less will this advantage become, until at some proportion an 
increase in distance will just pay for itself. As the proportionate 
length grows greater the advantage becomes a disadvantage 



§ 414. DISTANCE. 489 

until; when the competitive haul is entirely on the home road, 
any increase in distance becomes a net loss without any com- 
pensation. It is therefore advantageous for a road to be a 
short link in a long competitive route; an increase in that link 
will be financially advantageous; if the total length is less than 
that of the competing line, the advantage is still greater, for 
then the rate received per mile will be greater. 

413. Effect of the variations in the length of haul and the 
classes of the business actually done. The above distances 
refer to particular lengths of haul and are not necessarily the 
total lengths of the road. Each station on the road has 
traffic relations with an indefinite number of traffic points 
all over the country. The traffic between each station on 
the road and any other station in the country between which 
traffic may pass therefore furnishes a new combination, the 
effect of which will be an element in the total effect of a 
change of distance. In consequence of this, any exact solution 
of such a problem becomes impracticable, but a sufficiently 
accurate solution for all practical purposes is frequently ob- 
tainable. For it frequently happens that the great bulk of a 
road's business is non-competitive, or, on the other hand, it 
may be competitive-through, and that the proportion of one 
or more definite kinds of traffic is so large as to overshadow 
the other miscellaneous traffic. In such cases an approximate 
but sufficiently accurate solution is possible. 

414. General conclusions regarding a change in distance, 
(a) In all non-competitive business (local and through) the 
added distance is actually profitable. Sometimes practically 
all of the business of the road is non-competitive ; a considerable 
proportion of it is always non-competitive. 

(b) When the competitive local business is very large and the 
competitive through business has a very large average home 
haul compared with the foreign haul, the added distance is 
a source of loss. Such situations are unusual and are generally 
confined to trunk lines. 

(c) The above may be still further condensed to the general 
conclusion that there is always some compensation for the added 
cost of operating an added length of line and that it frequently 
is a source of actual profit. 

(d) There is, however, a limitation which should not be lost 
sight of. The above argument may be carried to the logical 



490 RAILROAD CONSTRUCTION. § 414. 

conclusion that, if added distance is profitable, the engineer 
should purposely lengthen the line. But added distance means 
added operating expenses. A sufficient tariff to meet these is a 
tax on the community — a tax which more or less discourages 
traffic. It is contrary to public policy to burden a community 
with an avoidable expense. But, on the other hand, a railroad 
is not a charitable organization, but a money-making enter- 
prise, and cannot be expected to unduly load up its first cost 
in order that subsequent operating expenses may be unduly 
cheapened and the tariff unduly lowered. A common reason 
for increased distance is the saving of the first cost of a very 
expensive although shorter line. 

(e) Finally, although there is a considerable and uncom- 
pensated loss resulting from curvature and grade which will 
justify a considerable expenditure to avoid them, there is by 
no means as much justification to incur additional expenditure 
to avoid distance. Of course needless lengthening should be 
avoided. A moderate expenditure to shorten the line may be 
justifiable, but large expenditures to decrease distance are 
never justifiable except when the great bulk of the traffic is 
exceedingly heavy and is competitive. 

415. Justification of decreasing distance to save time. It 
should be recalled that the changes which an engineer may 
make which are physically or financially possible will ordi- 
narily have but little effect on the time required for a trip. 
The time which can thus be saved will have practically no value 
for the freight business — at least any value which would justify 
changing the route. When there is a large directly competitive 
passenger traffic between two cities {e.g. New York to Phila- 
delphia) a difference of even 10 minutes in the time required 
for a run might have considerable financial importance, but 
such cases are comparatively rare. It may therefore be con- 
'cluded that the value of the time saved by shortening distance 
will not ordinarily be a justification for increased expense to 
accomplish it. 

416. Effect of change of distance on the business done. 
The above discussion is based on the assumption that the busi- 
ness done is unaffected by any proposed change in distance. 
If a proposed reduction in distance involves a loss of business 
obtained, it is almost certainly unwise. But if by increasing 
the distance the original cost of the road is decreased (because 



§ 416. DISTANCE. 491 

the construction is of less expensive character) , and if the receipts 
are greater, and are increased still more by an increase in busi- 
ness done, then the change is probably wise. While it is almost 
impossible in a subject of such complexity to give a general 
rule, the following is generally safe : Adopt a route of such length 
that the annual traffic per mile of line is a maximum. This 
statement may be improved by allowing the element of original 
cost to enter and say, adopt a route of such length that the annual 
traffic per mile of line divided by the average cost per mile is 
a maximum. Even in the above the operating cost per mile, 
as affected by the curvature and grades on the various routes, 
does not enter, but any attempt to formulate a general rule 
which would allow for variable operating expenses would e\d- 
dently be too complicated for practical application. 



CHAPTER XXII. 
CURVATURE. 

417. General objections to curvature. In the popular mind 
curvature is one of the most objectionable features of railroad 
alignment. The cause of this is plain. The objectionable 
qualities are on the surface, and are apparent to the non-tech- 
nical mind. They may be itemized as follows: 

1. Curvature increases operating expenses by increasing (a) 
the required tractive force, (b) the wear and tear of roadbed 
and track, (c) the wear and tear of equipment, and (d) the 
required number of track- walkers and watchmen. 

2. It may affect the operation of trains (a) by limiting the 
length of trains, and (b) by preventing the use of the heaviest 
tj^pes of engines. 

3. It may affect travel (a) bj^ the difficulty of making time, 
(b) on account of rough riding, and (c) on account of the appre- 
hension of danger. 

4. There is actually an increased danger of collision, derail- 
ment, or other form of accident. 

Some of these objections are quite definite and their true 
value may be computed. Others are more general and vague 
and are usually exaggerated. These objections will be dis- 
cussed in inverse order. 

418. Financial value of the danger of accident due to curva- 
ture. At the outset it should be realized that in general the 
problem is not one of curvature vs. no curvature, but simply 
sharp curvature vs. easier curvature (the central angle remain- 
ing the same), or a greater or less percentage of elimination 
of the degrees of central angle. A straight road between ter- 
mini is in general a financial (if not a physical) impossibility. 
The practical question is then, how much is the financial value 
of such diminution of danger that may result from such elimi- 
nations of curvature as an engineer is able to make? 

492 



§ 419. CURVATURE. 493 

In the year 1898 there were 2228 railroad accidents reported 
by the Railroad Gazette, whose Hsts of all accidents worth re- 
porting are very complete. Of these a very large proportion 
clearly had no relation whatever to curvature. But suppose 
we assume that 50% (or 1114 accidents) were directly caused 
by curvature. Since there are approximately 200,000 curves 
on the railroads of the country, there was on the average an 
accident for every 179 curves during the year. Therefore we 
may say, according to the theory of probabilities, that the 
chances are even that an accident may happen on any particular 
curve in 179 years. This assumes all curves to be equally danger- 
ous, which is not true, but we may temporarily consider it to be 
true. If, at the time of the construction of the road, $1.00 were 
placed at compound interest at 5% for 179 years, it would pro- 
duce in that time S620.89 for each dollar saved, wherewith to pay 
all damages, while the amount necessary to eliminate that cur- 
vature, even if it were possible, would probably be several thou- 
sand dollars. Tte number of passengers carried one mile for 
one killed in 1898-99 was 61,051,580. If a passenger were to 
ride continuously at the rate of sixty miles per hour, day and 
night, year after year, he would need to ride for more than 116 
years before he had covered such a mileage, and even then the 
probabilities of his death being due to curvature or to such a 
reduction of curvature as an engineer might accomplish are 
very small. Of course particular curves are often, for special 
reasons, a source of danger and justify the employment of 
special watchmen. They would also justify very large expen- 
ditures for their elimination if possible. But as a general 
proposition it is evidently impossible to assign a definite money 
value to the danger of a serious accident happening on a par- 
ticular curve which has no special elements of danger. 

Another element of safety on curved track is that trait of 
human nature to exercise greater care where the danger is more 
apparent. Many accidents are on record which have been 
caused by a carelessness of locomotive engineers on a straight 
track when the extra watchfulness usually observed on a curved 
track would hav^e avoided them. 

419. Effect of curvature on travel, (a) DifSculty in making 
time. The growing use of transition curves has largely elimi- 
nated the necessity for reducing speed on curves, and even when 
the speed is reduced it is done so easily and quickly by means 



494 RAILROAD CONSTRUCTION. § 419. 

of air-brakes that but little time is lost. If two parallel lines 
were competing sharply for passenger traffic, the handicap of 
sharp curvature on one road and easy curvature on the other 
might have a considerable financial value, but ordinarily the 
mere reduction of time due to sharp curvature will not have any 
computable financial value. 

(b) On account of rough riding. Again, this is much reduced 
by the use of transition curves. Some roads suffer from a gen- 
eral reputation for crookedness, but in such cases the excessive 
curvature is practically unavoidable. This cause probably 
does have some effect in influencing competitive passenger 
traffic. 

(c) On account of the apprehension of danger. This doubtless 
has its influence in deterring travel. The amount of its influence 
is hardly computable. When the track is in good condition 
and transition curves are used so that the riding is smooth, 
even the apprehension of danger will largely disappear. 

Travel is doubtless more or less affected by curvature, but 
it is impossible to say how much. Nevertheless the engineer 
should not ordinarily give this item any financial weight what- 
soever. Freight traffic (two thirds of the total) is unaffected 
by it. It chiefly affects that limited class of sharply competi- 
tive passenger traffic — a traffic of which most roads have not a 
trace. 

420. Effect on operation of trains, (a) Limiting the length 
of trains. When curvature actually limits the length of trains, 
as is sometimes true, the objection is valid aud serious. But 
this can generally be avoided. If a curve occurs on a ruling 
grade without a reduction of the grade sufficient to compensate 
for the curvature, then the resistance on that curve wifl be a 
maximum and that curve will limit the trains to even a less 
weight than that which may be hauled on the ruling grade. 
In such cases the unquestionably correct policy is to ^^com- 
pensate for curvature, '^ as explained later (see §§ 427, 428), and 
not allow such an objection to exist. It is possible for curvature 
to limit the length of trains even without the effect of grade. 
On the Hudson River R. R. the total net fall from Albany to 
New York is so small that it has practically no influence in 
determining grade. On the other hand, a considerable portion 
of the route follows a steep rocky river bank which is so crooked 
that much curvature is unavoidable and very sharp curvature 



§ 420. CURVATURE. 495 

can only be avoided by very large expenditure. As a consequence 
sharp curvature has been used and the resistance on the curves 
is far greater than that of any fluctuations of grade which it 
was necessary tc use. Or, at least, a comparatively small 
expenditure would siiiiice to cut do^\Ti any grade so that its 
resistance would be less than that of some curve which could 
not be avoided except at an enormous cost. And as a result, 
since the length of trains is really limited by ciu'vature, minor 
grades of 0.3 to 0.5% have been freely introduced which 
might be removed at comparatively small expense The above 
case is Yery unusual. Low grades are usually associated vdth. 
generally level country where curvature is easily avoided — 
as in the Camden and Atlantic R. R. Even in the extreme 
case of the Hudson River road the maximum curvature is 
only equivalent to a comparatively low ruling grade. 

(b) Preventing the use of the heaviest types of engines. The 
validity of this objection depends somewhat on the degree of 
curvature and the detailed construction of the engine. While 
some types of engines might have difficulty on curves of ex- 
tremely short radius, yet the objection is ordinarily invahd. 
This will best be appreciated when it is recalled that the " Con- 
solidation" type was originally designed for use on the sharp 
curvature of the mountain di^dsions of the Lehigh Valley R. R., 
and that the type has been found so satisfactory^ that it has 
been extensively employed elsewhere. It should also be re- 
membered that during the Ci\il War an immense traffic daily 
passed over a hastily constructed trestle near Petersburg, Va., 
the track having a radius of 50 feet. As a result of a test made 
at Renovo on the Philadelphia and Erie R. R. by Mr. Isaac 
Dripps, Gen. Mast. Mech., in 1875,* it was claimed that a 
Consolidation engine encountered less resistance per ton than 
one of the '^ American" type. Whether the test was strictly 
reliable or not, it certainly demonstrated that there was no 
trouble in using these heavy engines on very sharp curvature, 
and we may therefore consider that, except in the most extreme 
cases, this objection has no force whatsoever. 

* Seventh An. Rep. Am. Mast. Mech. Assn. 



496 



RAILROAD CONSTRUCTION. 



§421. 



EFFECT OF CURVATURE ON OPERATING EXPENSES. 

421. Relation of radius of curvature and of degrees of 
central angle to operating expenses. The smallest consideration 
will show that the sharper the curvature the greater will be 
the tractive force required, also the greater per unit of track 
length will be the rail wear and the general wear and tear on 
roadbed and rolling stock. But it would be inconvenient 
to use a relation between operating expenses and radius of 
curvature, because even when such a relation was found there 
would be two elements to consider in each problem — the radius 
and the length of the curve. The method which will be here 
developed cannot claim to be strictly accurate or even strictly 
logical, but, as will be shown later, the most uncertain elements 
of the computation have but a small influence on the final 
result, and the method is in general the only possible method of 
solution. The outline of the method is as follows: 

(1) For reasons given in detail later, it is found that the 
expenses, wear, etc., on the track from A to B will be substan- 
tially the same whether by the route M or N. The wear, etc., 




Fig. 208. 



per foot at N is of course greater, but the length of curve is 
less. Therefore the effect of the curvature depends on the 
degrees of central angle ^ and is independent of the radius. 



§ 422« CURVATURE. 497 

(2) At what degree of curvature is the total train resistance 
double its value on a tangent? Probably no one figure would 
be exact for all conditions. Train resistance varies with the 
velocity and with the various conditions of train loading even 
on a tangent, and it is by no means certain (or even probable) 
that the ratio would be exactly the same for all conditions. 
As an average figure we may say that a train running at average 
velocity on a 10° curve will encounter a resistance due to cur- 
vature which is approximately equal to the average resistance 
found on a level tangent. On a 10° curve therefore the resist^ 
ance is doubled. 

(3) A train-mile costs about so much — approximately $1.35. 
Doubling the tractive resistance will increase certain items oX 
expenditure about so much. Their combined value is so much 
'per cent of the cost of a train-mile. A mile of continuous 10^ 
curve contains 528° of central angle. A mile of such track 
would add so much per cent to the average train -mile expenses, 
and each degree of central angle is responsible for g-^-g- of this 
increase. Since the increase is irrespective of radius and de* 
pends only on the degrees of central angle, we therefore saj'' 
that each degree of central angle of a curve T\dll add so much 
to the average operating expenses of a train-mile. 

The "cost per train-mile" considered above should be con- 
sidered as the cost of a mile of level tangent. If w^e for a moment 
consider that all the railroads of the country were made abso- 
lutely straight and level, it is apparent that the average cost 
per train-mile instead of being about $1.35, would be somewhat 
less. The percentage should therefore be applied to this reduced 
value, but the net effect of this change would evidentl}^ be 
small. I 

422. Effect of curvature on maintenance of way. A 
very large proportion of the items of expense in a train-mile 
are absolutely unaffected by curvature. It will therefore 
simplify matters somewhat if we at once throw out all the un- 
affected items. Of the items of maintenance of way and struc- 
tures all but items 2 to 6 may be thrown out. Item 9 will be 
somewhat affected when bridges or trestles occur on a curve. 
But when it is considered what a very small percentage of this 
small item (1.709%) could be ascribed to curvature, since the 
very large majority of bridges and trestles are purposely made 



498 RAILROAD CONSTRUCTION. § 422. 

straight, and since culverts, etc., are not affected, we may 
evidently ignore any variation in the item. 

Item 3. Renewals of Ties. Curvature affects ties by in- 
creasing the ''rail cutting '^ and on account of the more frequent 
respiking, which ''spike-kills'' the ties even before they have 
decayed. Wellington estimates that a tie which will last nine 
years on a tangent will last but six years on a 10° curve. He 
adds 50% for tie renew^als. He considers the decrease in tie 
life to be proportional to the degree of curve and therefore 
again verifies the general statement made above regarding the 
expense of curvature. 

Item 4. Renewals of Rails. The excess wear due to curv- 
ature has never been determined with satisfactory conclu- 
siveness. Some tests have been made within the last few years 
on the Northern Pacific Railroad, which have perhaps followed 
the only practical method for determining the law of rail wear 
on curves. Selected rails on several tangents and curves of 
varying degrees of curvature were annually taken up, cleaned 
and weighed, and the annual loss due to wear was noted. The 
results indicated a loss of w^eight on curves varying nearly 
according to the degree of curve, and that the excess wear on 
a curve is 22.6% per degree of curve over that on a tangent. 
For a 10° curve, this would mean an excess wear of 226%. 

Item 6. Repairs of Roadway. A very large proportion of 
the subitems are absolutely unaffected. The care of embank- 
ments and slopes, the ditching, weeding, etc., are evidently 
unaffected. The track-labor on rails and ties and the work 
of surfacing will evidently be somewhat increased, and yet 
it is very seldom that the length of a track section would be de- 
creased simply on account of excessive curvature throughout 
that section. We are here trying to estimate how much this 
item, which consists largely of track-labor, will be affected by 
528° of central angle per mile. In the previous chapter an 
, approximate estimate was made that the average curvature 
per mile of road for the whole United States is about 35°. 528° 
of curvature in a mile probably does not frequently occur. 
It would mean the equivalent of nearly IJ complete circles, 
and yet it is probably a generous estimate to say that the track- 
labor and other expenses belonging to this item would not be 
increased more than 25% for such an amount of curvature. 
Items 2 and 5 are also allowed 25%. 



t 



§ 423. CURVATURE. 499 

423. Effect of curvature on mainten^ce of equipment. All 

items except the repairs, renewals and depreciation of steam 
locomotives, passenger-, freight- and work-cars, and shop 
machinery and tools, will be considered as unaffected. As 
before, electric equipment is ignored. 

Items 25-27. Repairs and renewals of locomotives. Curva- 
ture affects locomotive repairs by increasing very largely the 
wear on tires and wheels, and also the wear and strain due to 
the additional power required. Since the resistance due to 
curvature is very small compared with that due to even a mod- 
erate grade, this last cause may be neglected altogether. Re- 
ferring to the tabulation in § 406, we find an estimate that 19% 
of the cost of engine repairs is assigned to curvature and grades 
combined. Of this amount two-thirds, or, say, 13%, should 
be assigned to curvature alone. On the basis that the average 
curvature of the roads of the country is about 35° per mile, 
which is about one-fifteenth of the 528° of curvature per mile 
which we are considering, then, if 35° is responsible for 13% 
increase in engine repairs, 528° would be responsible for 196%. 
It must be admitted that the above computation is grossly 
approximate, and that it contains the unwarrantable assump- 
tion that the extra cost of engine repairs which is due to curva- 
ture will be strictly in proportion to the degrees of curve. 
Although it is probably not true that 528° of curvature would 
increase the cost of engine repairs by 15 times the extra cost 
of 35° of curvature, yet it is probably true that for ordinary 
variations from that average of 35° per mile the increased cost 
of engine repairs will be approximately as the number of degrees 
of curve, and therefore our final value is not necessarily far out 
of the way. If 35° is responsible for an increase of 13%, 1° 
would be responsible for about .37 of 1%. In allowing an 
increase of 196% for 528° we are also allowing .37 of 1% per 
degree of central angle. 

Items 31-36, 43-45. By a similar course of reasoning to 
that above given, the estimates for Items 34-36 and 43-45 will 
be made 100%, while that for Items 31-33 will be made only 
50%, because such a large proportion of the expenses of Items 
31-33 are due to painting and maintaining upholstery, which 
have no relation to variations in ahnement. 

Item 46. The repairs and renewals of shop machinery 
and tools will not be increased more than 50% per mile 



500 RAILROAD CONSTRUCTION. § 423. 

for the additional repairs required on the above equip- 
ment. 

424. Effect of curvature on conducting transportation. An 

inspection of the items under this general heading will show 
us Uiat we may at once throw out as unaffected all the items 
except those which concern engine-suppHes for road engines, 
flagmen, and watchmen, and the group* which refers to 
accidents. 

Items 82, 83, 84, and 85. The required quantities in this 
calculation are the additions to cost resulting from the intro- 
duction of 528° of central angle into a mile of track. We have 
assumed that this amount of curvature will exactly double the 
resistance. We found in Chapter XXI (§ 407) that the average 
increase in fuel consumption due to direct hauling amounts to 
about 44%. We have here assumed that the added curvature 
exactly doubles the work. We will therefore charge 44% of 
the average cost of a train-mile for this extra curvature. Since 
the consumption of water and other engine-supplies is roughly 
proportional to the consumption of coal, there will evidently 
be no error worth considering in assigning this same percentage 
to Items 83, 84, and 85. 

Flagmen. There are many cases where a dangerous curve 
justifies and requires the employment of a special flagman to 
give timely notice of any dangerous condition of the track. 
Such special cases would, of course, justify a considerable expend- 
iture to eliminate the dangerous features of that particular 
location, but such a charge should not be made against curvature 
in general. Ordinarily the elimination or retention of a curve 
will not involve the question of watchmen and flagmen in any 
way. We are therefore justified in disregarding this item 
altogether as a general proposition, if we keep in mind that the 
item should be included when we are considering the ehmina- 
tion of some particularly dangerous curve. 

Items 99, 99-103. This group of items, which refer to 
accidents and the increased danger of accident due to curvature, 
and therefore the amount of money which might be justifiably 
spent to avoid this danger, has already been discussed in § 418. 
It was there shown that, although there might be special cases 
which would justify considerable expenditure on account of 
specific dangers in the situation, we cannot ordinarily give any 
definite financial value to this item as appHed to curvature in 



§425. 



CURVATURE. 



501 



general. We therefore must eliminate these items as affecting 
the cost of curvature in general. 

General expenses. Items 106 to 116 will also be unaffected. 

425. Estimate of total effect per degree of central angle. 
CompiUng the above estimates we have Table XXII. Accord- 

TABLE XXII. EFFECT ON OPERATING EXPENSES OF CHANGES 
IN CURVATURE. 



Item 
number. 


Item (abbreviated). 


Normal 
average. 


Per cent 
affected. 


Cost per 

mile, 
per cent. 


2 
3 


Ballast 

Ties 


0.50 
3.10 
0.92 
1.13 
7.53 
6.91 


25 
50 
226 
25 
25 



0.12 
1.55 


4 


Rails . 


2.10 


5 


Otber track m^aterial 


28 


6 


Road^^ay and track 


1.88 




(All othier item^s) 







Maintenance of way 






20.09 




5.93 









25-27 


Locomotives 


8.61 
2.14 
10.15 
0.34 
0.53 
0.97 


196 

50 

100 

100 

50 




16.88 


31-33 


Passenger-cars 


1 07 


34-36 


Freight-cars 


10.15 


43-45 


W^ork-cars 


34 


46 


Shop machinery; tools 

(All other items) 


0.26 





Maintenance of equipment 






22.74 




28.70 








53-60 


Traffic 


3.08 














82-85 


Fuel and other supplies for road 
engines . . 


11.41 
39.03 


44 



5 02 




(All other items) 







Transportation *. * 






50.44 




5 02 










106-116 


General expenses 


3.65 


















100.00 




39 65 









ing to the table 528° of curvature in one mile will increase the 
expenses of each train passing over it by 39.65% of the average 
cost of a train-mile, and, according to the general principles 
laid down in § 425, one degree of central angle of any curve, no 

matter what the radius, will increase the expenses by of 

528 

39.65%, or .0751% per degree. Therefore, the cost per year per 
daily train each way, at the average rate of $1.50 per train- 
mile would be 

150X.0751%X2X365-82.23 c. 

For a round number we will call this 82 cents. 



502 



EAILBOAD CONSTRUCTION. 



§425. 



To forestall one kind of objection to the foregoing course 
of reasoning, it should be remembered that many of the estimates 
of additional cost, instead of being actually based. on the effects 
of a continuous 10° curve, were based on the observed effects 
of lighter curvature, which were then multiplied by a factor 
to obtain the effect of a continuous 10° curve, as if the effects 
of curvature were strictly proportional to the curvature; but 
since we afterward divide our final result by 528 to obtain the 
effect of one degree of curvature, and then multiply this constant 
by the number of degrees of central angle, as found in ordinary 
practice, our calculations are not thereby vitiated because 
the effect of curvature is not strictly proportional to the rate 
of curvature. It is probably true that within the ordinary 
limits of variations in rate of curvature the above calculations 
are substantially true. In extreme cases they are probably 
in error, although it is likewise probably true that extreme 
curvature will have a variable effect on the rate of increase of 
the various items, and that even in extreme cases the error will 
not be very large. 




Fig. 209. 



As a simple illustration (a more extended one will be given 
later), suppose that by using greater freedom with regard to 



§ 426. CURVATURE. 503 

earthwork "the crooked line sketched may be reduced to the 
shnple curve shown and a curvatiu-e of, say, 110° may be re- 
duced to, say, 60°. 

Note that since the extreme tangents are identical, the sav- 
ing in central angle results from the elimination of the reversed 
curvature and of that part of the direct curvature necessary to 
balance the reversed curvature. Assume that there are six 
daily trains each way. Then^the annual saving is 

50 X. 82X6 =$246, 

which at 5% would justify an expenditure of $4920. If 
the extra cost of construction does not exceed this, the im- 
provement is justifiable, and is made all the more so if the proba- 
bilities are great that the future traffic will largely exceed six 
trains per day. At the same time the warning regarding '' dis- 
counting the future" with respect to expected traffic should 
not be neglected. The possible effect of change of distance 
has not been referred to in the above problem. In any case it 
is a distinct problem. According to the above sketch, the 
difference in distance is probabl}^ very slight, and consider- 
ing the compensating character of extra distance, such small 
differences may usually be disregarded. The possible effect 
of change of grade will be discussed in the next chapter. As- 
suming that there is no difference to be considered on account 
of either grade or distance, the question hinges on the advisa- 
bility of spending $4920 for the improvement. 

426. Reliability and value of the above estimate. It should 
be realized at the outset that no extreme accuracy is claimed 
for the above estimate. The effect of curvature is somewhat 
variable as well as uncertain, but such estimates have this great 
value. Vary the estimates of indi^ddual items as you please 
(T\athin reason), and the final result is still about the same and 
may be used to guide the judgment. As an illustration, sup- 
pose that the item of renewals of rails is assumed to be affected 
300% rather than 226%, the justifiable expenditure to avoid 
the curvature in the above case may similarly be computed 
as $5017, an increase of about 2%. But, after all, the real 
question is not whether the improvement is worth $4800 or 
$5017. The extra work involved may perhaps be done for $500 
or it may require $10000. The above general method furnishes 



504 RAILROAD CONSTRUCTION. § 426. 

reliance on vague judgment that it should not be ignored. 

COMPENSATION FOR CURVATURE. 

427. Reasons for compensation. The effect of curvature on 
a grade is to increase the resistance by an amount which is equiv- 
alent to a material addition to that grade. On minor grades 
the addition is of little importance, but when the grade is nearly 
or quite the ruling grade of the road, then the additional resist- 
ance induced by a curve will make that curve a place of maxi- 
mum resistance and the real maximum will be a 'Virtual grade '* 
somewhat higher than the nominal maximum. If, in Fig. 210, 

Tang.__ ^ 



Fig. 210. 

AN represents an actual uniform grade consisting of tangents 
and curves, the ^^ virtual grade'' on curves at BC and DE may 
be represented by BC and DE. If BC and DE are very long, 
or if a stop becomes necessary on the curve, then the full dis- 
advantage of the curve becomes developed. If the whole grade 
may be operated without stoppage, then, as elaborated further 
in the next chapter, the whole grade may be operated as if equal 
to the average grade, AF, which is better than J5(7, although 
much worse than AN. The process of '^compensation" con- 
sists in reducing the grade on every curve by such an amount 
that the actual resistance on each curve, due to both curvature 
and grade, shall precisely equal the resistance on the tangent. 
The practical effect of such reduction is that the '^ virtual" grade 
is kept constant, while the nominal grade fluctuates. 

One effect of this is that (see Fig. 211) instead of accomplish- 
ing the vertical rise from A to 6^ (i.e., HG) in the horizontal 
distance AHj it requires the horizontal distance AK. Such an 
addition to the horizontal distance can usually be obtained by 
proper development, and it should always be done on a ruling 



§428. 



CURVATURE. 



505 



grade. Of course it is possible that it will cost more to accom- 
plish this than it is worth, but the engineer should be sure of 
this before allowing this virtual increase of the grade. 




Fig. 211. 



European engineers early realized the significance of unre- 
duced curvature and the folly of laying out a uniform ruling 
grade regardless of the curvature encountered. Curve compen- 
sation is now quite generally allowed for in this country, but 
thousands of miles have been laid out without any compensa- 
tion. A very common limitation of curvature and grade has 
been the alliterative figures 6° curvature and 60 feet per mile 
of grade, either singly or in combination. Assuming that the 
resistance on a 6° curve is equivalent to a 0.3% grade (15.84 feet 
per mile), then a 6° curve occurring on a 60-foot grade would 
develop more resistance than a 75-foot grade on a tangent. 
The '^mountain cut-off'' of the Lehigh Valley Railroad near 
Wilkesbarre is a fine example of a heavy grade compensated 
for curvature, and yet so laid out that the virtual grade is imi- 
form from bottom to top, a distance of several miles. 

428. The proper rate of compensation. This evidently is the 
rate of grade of which the resistance just equals the resistance 
due to the curve. But such resistance is variable. It is greater 
as the velocity is lower ; it is generally about 2 lbs. per ton 
(equivalent to a 0.1% grade) per degree of curve when starting 
a train. On this account, the compensation for a curve which 
occurs at a known stopping-place for the heaviest trains should 
be 0.1% per degree of curve. The resistance is not even strictly 
proportional to the degree of curvature, although it is usually 
considered to be so. In fact most formulae for curve resistance 
are based on such a relation. But if the experimentally deter- 
mined resistances for low curvatures are applied to the excessive 



506 RAILROAD CONSTRUCTION. § 428. 

curvature of the New York Elevated road, for exalnple, the 
rules become ridiculous. On this account the compensation 
per degree of curve may be made less on a sharp curve than on 
an easy curve. The compensation actually required for very 
fast trains is less than for slow trains, say 0.02 or 0.03% per 
degree of curve; but since the comparatively slow and heavy 
freight trains are the trains which are chiefly limited by ruhng 
grade, the compensation must be made with respect to those 
trains. From 0.04 to 0.05% per degree is the rate of compen- 
sation most usually employed for average conditions. Curves 
which occur below a known stopping-place for all trains need 
not be compensated, for the extra resistance of the curve will 
be simply utilized in place of brakes to stop the train. If a curve 
occurs just above a stopping-place, it is very serious and should 
be amply compensated. Of course the down-grade traffic need 
not be considered. 

It sometimes happens that the ordinary rate of compensa- 
tion will consume so much of the vertical height (especially if 
the curvature is excessive) that a steeper through grade must 
be adopted than was first computed, and then the trains might 
stall on the tangents rather than on the curves. In such cases 
a slight reduction in the rate of compensation might be justi- 
fiable. The proper rate of compensation can therefore be 
estimated from the following rules : 

(1) On the upper side of a stopping-place for the heaviest 
trains compensate 0.10% per degree of curve. 

(2) On the lower side of such a stopping-place do not com- 
pensate at all. 

(3) Ordinarily compensate about 0.05% per degree of curve. 

(4) Reduce this rate to 0.04% or even 0.03% per degree 
of curve if the grade on tangents must be increased to reach 
the required summit. 

(5) Reduce the rate somewhat for curvature above 8° or 10°. 

(6) Curves on minor grades need not be compensated, unless 
the minor grade is so heavy that the added resistance of the 
curve would make the total resistance greater than that of the 
the ruling grade, or unless there is some ground to believe that 
the ruling grade may sometime be reduced below that of the 
minor grade under consideration. 

429. The limitations of maximum curvature. What is the 
maximum degree of curvature which should be allowed on any 



§ 429. CURVATURE. 507 

road? It has been shown that sharp curvature does not prevent 
the use of the heaviest types of engines, and although a sharp 
curve unquestionably increases operating expenses, the increase 
is but one of deo;ree with hardly any definite limit. The general 
character of the country and the gross capital available (or 
the probable earnings) are generally the true criterions. 

A portion of the road from Denver to Leadville, Col., is an 
example of the necessity of considering sharp curvature. The 
traffic that might be expected on the line was so meagre and 
yet the general character of the country was so forbidding 
that a road built according to the usual standards would have 
cost very much more than the traffic could possibly pay for. 
The line as adopted cost about $20,000 per mile, and yet in a 
stretch of 11.2 miles there are about 127. curves. One is a 25° 
20' curve, twenty-four are 24° curves, twenty-five are 20° curves, 
and seventy- two are sharper than 10°. If 10° had been made 
the limit (a rather high limit according to usual ideas), it is 
probable that the line would have been found impracticable 
(except with prohibitive grades) unless four or five times as 
much per mile had been spent on it, and this would have ruined 
the project financially. 

For many years the main-line trafl^ic of the Baltimore and 
Ohio R. R. has passed over a 300-foot curve (19° 10') and a 
400-foot curve (14° 22') at Harper's Ferry. A few years ago 
some reduction was made in this by means of a tunnel, but 
the fact that such a road thought it wise to construct and operate 
such curves (and such illustrations on the heaviest-traffic roads 
are quite common) shows how foolish it is for an engineer to 
sacrifice money or (which is much more common) sacrifice 
gradients in order to reduce the rate of curvature on a road 
which at its best is but a second- or third-class road. 

Of course such belittling of the effects of curvature may 
be (and sometimes is) carried to an extreme and cause an engi- 
neer to fail to give to curvature its due consideration. Degrees 
of central angle should always be reduced by all the ingenuity 
of the engineer, and should only be limited by the general rela- 
tion between the financial and topographical conditions of the 
problem. Easy curvature is in general better than sharp curva- 
ture and should be adopted when it may be done at a small 
financial sacrifice, especially since it reduces distance generally 
and may even cut down the initial cost of that section of the 



508 RAILROAD CONSTRUCTION. § 429. 

road. But large financial expenditures are rarely, if ever, jus- 
tifiable where the net result is a mere increase in radius v/ithout 
a reduction in central angle. An analysis of the changes which 
have been so extensively made during late years on the Penn. 
R. R. and the N. Y., N. H. & H. R. R. will show invariably a 
reduction of distance, or of central angle, or both, and perhaps 
incidentally an increase in radius of curvature. There are but 
few, if any, cases where the sole object to be attained by the 
improvement is a mere increase in radius. 

The requirements of standard M. C. B. car-couplers have 
virtually placed a limitation on the radius on account of the 
corners of adjacent cars striking each other on very sharp 
curves. This limitation has been crystallized into a rule on 
the P. R. R. that no. curve, even that of a siding, can have a 
less radius than 175 feet, which is nearly the radius of a 33° 
curve. Of course only the most peremptory requirements of 
yard work would justify the employment of such a radius. 



1 



CHAPTER XXIII. 

GRADE. 

430. Two distinct effects of grade. The effects of grade on 
train expenses are of two distinct kinds; one possible effect is 
very costly and should be limited even at considerable expen- 
diture* the other is of comparatively little importance, its cost 
being slight. As long as the length of the train is not limited, 
the occurrence of a grade on a road simply means that the engine 
is required to develop so many foot-pounds of work in raising 
the train so many feet of vertical height. For example, if a 
freight train weighing 600 tons (1,200,000 lbs.) climbs a hill 
50 feet high, the engine performs an additional work of creating 
60,000,000 foot-pounds of potential energy. If this height is 
surmounted in 2 miles and in 6 minutes of actual time (20 
miles per hour), the extra work is 10,000,000 foot-pounds per 
minute, or about 303 horse-power. But the disadvantages of 
such a rise are always largely compensated. Except for the fact 
that one terminus of a road is generally higher than the other, 
every up grade is followed, more or less directly, by a down grade 
which is operated partly by the potential energy acquired during 
the pre\aous climb. But when w^e consider the trains running 
in both directions even the difference of elevation of the termini 
is largely neutralized. If we could eliminate frictional resist- 
ances and particularly the use of brakes, the net effect of minor 
grades on the operation of minor grades in both directions would 
be zero. Whatever was lost on any up grade would be regained 
on a succeeding down grade, or at any rate on the return trip. 
On the very lowest grades (the limits of which are defined later) 
we may consider this to be literally true, viz., that nothing is 
lost by their presence; whatever is temporarily lost in climbing 
them is either immediately regained on a subsequent light down 
grade or is regained on the return trip. If a stop is required 
at the bottom of a sag, there is a net and uncompensated loss 

of energy. 

509 



510 RAILROAD CONSTRUCTION. § 430. 

On the other hand, if the length of trains is limited by the 
grade, it will require more trains to handle a given traffic. The 
receipts from the traffic are a definite sum. The cost of hand- 
ling it will be nearly in proportion to the number of trains. 
Anticipating a more complete discussion, it may be said as an 
example .that increasing the ruling grade from 1.20% (63.36 
feet per mile) to 1.55% (81.84 feet per mile — an increase of 
about 18.5 feet per mile) will be sufficient to increase the re- 
quired number of trains for a given gross traffic about 25%, 
i.e., five trains will be required to handle the traffic which four 
trains would have handled before at a cost slightly more than 
four-fifths as much. The effect of this on dividends may readily 
be imagined. 

431. Application to the movement of trains of the laws of 
accelerated motion. When a train starts from rest and acquires 
its normal velocity, it overcomes not only the usual tangent 
resistances (and perhaps curve and grade resistances), but it 
also performs work in storing into the train a vast fund of kinetic 
energy. This work is not lost, for every foot-pound of such 
energy may later be utilized in overcoming resistances, pro- 
vided it is not wasted by the action of train-brakes. If for a 
moment we consider that a train runs without any friction, 
then, when running at a velocity of v feet per second, it possesses 
a kinetic energy which would raise it to a height h feet, when 

h = ^r-, in which g is the acceleration of gravity = 32.16. Assuni- 

ing that the engine is exerting just enough energy to overcome 
the frictional resistances, the train would climb a grade until the 
train was raised h feet above the point where its velocity was v. 
When it had climbed a height h^ (less than h) it would have 
velocity Vi=\/2g{h — h'). As a numerical illustration, assume 

v = 30 miles per hour = 44 feet per second. Then /^ = — = 30. 1 feet, 

and assuming that the engine was exerting just enough force 
to overcome the rolling resistances on a level, the kinetic 
energy in the train would carry it for two miles up a grade of 
15 feet per mile, or half a mile up a grade of 60 feet per mile. 
When the train had climbed 20 feet, there would still be 10.1 
feet left and its velocity would be ?;i=\/2(7(10.1) =25.49 feet 
per second = 17.4 miles per hour. These figures, however, must 
be slightly modified on account of the weight and the revolving 



§ 432. GRADE. 511 

action of the wheels, which form a considerable percentage 
of the total weight of the train. When train velocity is being 
acquired, part of the work done is spent in imparting the energy 
of rotation to the driving-wheels and various truck-wheels of 
the train. Since these wheels run on the rails and must turn 
as the train moves, their rotative kinetic energy is just as effec- 
tive — as far as it goes — in becoming transformed back into 
useful work. The proportion of this energy to the total kinetic 
energy has already been demonstrated (see Chapter XVI, 
§ 347). The value of this correction is variable, but an average 
value of 5% has been adopted for use in the accompanying 
tabular form (Table XXIII), in which is given the corrected 
'S^elocit}^ head" corresponding to various velocities in miles 
per hour. The table is computed from the following formula : 

^r 1 '^ 1 1 t'Mn ft. per sec. 2.151'y^inm.perh. ^^oo^^ 2 
Velocity head = ^ ^ = 64.32 =0.03344.^ 

adding 5% for the rotative kinetic energy of the wheels, 0.00167^'^ 



The corrected velocit}^ head therefore equals 0.03511 y^ 

Part of the figures of Table XXIII were obtained by inter- 
polation and the final hundredth may be in error by one unit, 
but it may readily be shown that the final hundredth is of no 
practical importance. It is also true that the chief use made 
of this table is with velocities much less than 50 miles per hour. 
Corresponding figures may be obtained for higher velocities, if 
desired, by multiplying the figure for half the velocity by four. 

432. Construction of a virtual profile. The following simple 
demonstration will be made on the basis that the ordinary 
tractive resistances and also the tractive force of the locomo- 
tive are independent of velocity. For a considerable range of 
velocity which includes the most common freight-train velocities 
this assumption is so nearly correct that the method will give 
an approximately correct result, but for higher velocities and 
for more accurate results a more complicated method (given 
later) must be used. The following demonstration will serve 
well as a preliminary to the more accurate method. It may 
best be illustrated by considering a simple numerical example. 

Assuming that a train is passing A (see Fig. 212), running at 
30 miles per hour. Assume that the throttle is not changed or 
any brakes applied, but that the engine continues to exert the 



512 



EAILROAD CONSTRTJCTION. 



§432. 



table xxiii. — velocity head (representing the kinetic 
energy) of trains moving at various velocities. 



Vel. 






















mi. 
hr. 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


10 


3.51 


3.58 


3.65 


3.72 


3.79 


3.87 


3.95 


4.02 


4.10 


4.17 


11 


4.25 


4.33 


4.41 


4.49 


4.57 


4.65 


4.73 


4.81 


4.89 


4.97 


12 


5.06 


5„15 


5.23 


5.32 


5.41 


5.50 


5.58 


5.67 


5.75 


5.84 


13 


5.93 


6.02 


6.12 


6.21 


6.31 


6.40 


6.50 


6.59 


6.69 


6.78 


14 


6.88 


6.98 


7.08 


7.19 


7.29 


7.39 


7.49 


7.60 


7.70 


7.80 


15 


7.90 


8.00 


8.11 


8.22 


8.33 


8.44 


8.55 


8.66 


8.77 


8.88 


16 


8.99 


9.10 


9.21 


9.32 


9.43 


9.55 


9.67 


9.79 


9.91 


10.03 


17 


10.15 


10.27 


10.39 


10.51 


10.63 


10.75 


10.87 


10.99 


11.12 


11.25 


18 


11.38 


11.50 


11.63 


11.76 


11.89 


12.02 


12.15 


12.28 


12.41 


12.55 


19 


12.68 


12.81 


12.95 


13.08 


13.22 


13.35 


13.49 


13.63 


13.77 


13.91 


20 


14.05 


14.19 


14.33 


14.47 


14.61 


14.75 


14.89 


15.04 


15.19 


15.34 


21 


15.49 


15.64 


15.79 


15.94 


16.09 


16.24 


16.39 


16.54 


16.69 


16.84 


22 


17.00 


17.15 


17.30 


17.46 


17.62 


17.78 


17.94 


18.10 


18.26 


18.42 


23 


18.58 


18.74 


18.90 


19.06 


19.22 


19.38 


19.55 


19.72 


19.89 


20.06 


24 


20.23 


20.40 


20.57 


20.74 


20.91 


21.08 


21.25 


21.42 


21.59 


21.77 


25 


21.95 


22.12 


22.30 


22.48 


22.66 


22.84 


23.02 


23.20 


23.38 


23.56 


26 


23.74 


23.92 


24.10 


24.28 


24.46 


24.65 


24.84 


25.03 


25 22 


25.41 


27 


25.60 


25.79 


25.98 


26.17 


26.36 


26.55 


26.74 


26.93 


27.13 


27.33 


28 


27.53 


27.73 


27.93 


28.13 


28.33 


28.53 


28.73 


28.93 


29.13 


29.33 


29 


29.53 


29.73 


29.93 


30.13 


30.34 


30.55 


30.76 


30.97 


31.18 


31.39 


30 


31.60 


31.81 


32.02 


32.23 


32.44 


32.65 


32.86 


33.08 


33.30 


33.52 


31 


33.74 


33.96 


34.18 


34.40 


34.62 


34.84 


35.06 


35.28 


35.50 


35.72 


32 


35.95 


36.17 


36.39 


36.62 


36.85 


37.08 


37.31 


37.54 


37.77 


38.00 


33 


38.23 


38.46 


38.69 


38.92 


39.15 


39.38 


39.62 


39.86 


40.10 


40.34 


34 


40.58 


40.82 


41.06 


41.30 


41.54 


41.78 


42.02 


42.26 


42.51 


42.76 


35 


43.01 


43.26 


43.51 


43.76 


44.01 


44.26 


44.51 


44.76 


45.01 


45.26 


36 


45.51 


45.76 


46.01 


46.26 


46.52 


46.78 


47.04 


47.30 


47.56 


47.82 


37 


48.08 


48.34 


48.60 


48.86 


49.12 


49.38 


49.64 


49.91 


50.18 


50.45 


38 


50.72 


50.99 


51.26 


51.53 


51.80 


52.07 


52.34 


52.61 


52.88 


53.15 


39 


53.42 


53.69 


53.96 


54.23 


54.51 


54.79 


55.07 


55.35 


55.63 


55.91 


40 


56.19 


56.47 


56.75 


57.03 


57.31 


57.59 


57.87 


58.16 


58.45 


58.74 


41 


59.03 


59.32 


59.61 


59.90 


60.19 


60.48 


60.77 


61.06 


61.35 


61.64 


42 


61.94 


62.23 


62.52 


62.82 


63.12 


63.42 


63.72 


64.02 


64.32 


64.62 


43 


64.92 


65.22 


65.52 


65.82 


66.12 


66.43 


66.74 


67.05 


67.36 


67.67 


44 


67.98 


68.29 


68.60 


68.91 


69.22 


69.53 


69.84 


70.15 


70.46 


70.78 


45 


71.10 


71.42 


71.74 


72.06 


72.38 


72.70 


73.02 


73.34 


73.66 


73.98 


46 


74.30 


74.62 


74.94 


75.26 


75.59 


75.92 


76.25 


76.58 


76.91 


77.24 


47 


77.57 


77.90 


78.23 


78.56 


78.89 


79.22 


79.55 


79.89 


80.23 


80.57 


48 


80.91 


81.25 


81.59 


81.93 


82.27 


82.61 


82.95 


83 29 


83.63 


83.97 


49 


84.32 84.66 


85.00 


85.34 


85.69 


86.04 


86.39 


86.74 


87.09 


87.44 


50 


87.79 


88.14 


88.49 


88.85 


89.20 


89.55 


89.91 


90.26 


90.61 


90.97 



same draw-bar pull. At A its ^S^elocity head^' is that due to 30 
miles per hour, or 31.60 feet. At B it has gained 40 feet more, 
and its velocity is that due to a velocity head of 71.60 feet, or 
slightly over 45 miles per hour. At B^ its velocity is again 30 i 
miles per hour and velocity head 31.60 feet. At C the velocity 
head is but 6.60 feet and the velocity about 13.7 miles per hour. 



§432. 



GRADE. 



513 



As the train runs from C to D its Velocity increases to 30 miles at 
C and to over 45 miles per hour at D. At E the velocity again 
becomes 30 miles per hour. Although there will be some slight 
modifications of the above figures in actual practice, yet the above 
is not a fanciful theoretical sketch. Thousands of just such 
undulations of grade are daily operated in such a way, without 
disturbing the throttle or applying brakes, and the draw-bar 
.pull, if measured by a dynamometer, would be found to be 
i practically constant. Of course the above case assimies that 

^rtuaT proffle 




Fig. 212. 
there are no stoppages and that the speed through the sags is 
not so great that safety requires the application of brakes. 
Observe that the ^'virtual profile" is here a straight line — as it 
always is when the draw-bar pull is constant. The virtual 
profile (in this case as well as in every other case, illustrations 
of which vAW follow) is found by adding to the actual profile 
at any point an ordinate which represents the ^'velocity head" 
due to the velocity of the train at that point. 

As another case, assume that a train is climbing the grade AE 
and exerting a pull just sufficient to maintain a constant velocity 
p / up that grade. Then A^B^ (parallel 
to AB) is the virtual profile, A A' 
representing the velocity head. A 
stop being required at C, steam is 
shut off and brakes are applied 
at Bj and the velocity he^d BB^ 
reduces to zero at C. The train 
starts from C, and at D attains a velocity corresponding to 
the ordinate DD\ At D the throttle may be shghtly closed 
so that the velocity will be uniform and the virtual grade is 
D'E% parallel to DE. 

From the above it may be seen that a virtual profile has the 
following properties: 

(a) When the velocity is uniform, the virtual profile is parallel 
with the actual. 




Fig. 213. 



514 RAILROAD CONSTRUCTION. § 432. 

(b) When the velocity is increasing the profiles are separating; 
when decreasing the profiles are approaching. 

(c) When the velocity is zero the profiles coincide. 

433. Use, value, and possible misuse. The essential feature 
respecting grades is the demand on the locomotive. From the 
foregoing it may readily be seen that the ruling grade of a road 
is not necessarily the steepest nominal grade. When a grade 
may be operated by momentum, i.e., when every train has an 
opportunity to take ^'sl run at the hill,'' it may become a very 
harmless grade and not limit the length of trains, while another 
grade, actually much less, which occurs at a stopping-place 
for the heaviest trains, will require such extra exertion to get 
trains started that it may be the worst place on the road. There- 
fore the true way to consider the value of the grade at any criti- 
cal place on the road is to construct a virtual profile for that 
section of the road. The required length of such a profile is 
variable, but in general may be said to be limited by points on 
each side of the critical section at which the velocity is definite, 
as at a stopping-place (velocity zero), or a long heavy grade where 
it is the minimum permissible, say 10 or 15 miles per hour. 

Since the velocities of different trains vary, each train will 
have its own virtual profile at any particular place. The fast 
passenger trains are generally unaffected, practically. The 
requirement of high average speed necessitates the use of power- 
ful engines, and grades which would stall a heavy freight will 
only cause a momentary and harmless reduction of speed of 
the fast passenger train. 

A possible misuse of virtual profiles lies in the chance that a 
station or railroad grade crossing may be subsequently located 
on a heavy grade that was designed to be operated by momen-j 
tum. But this should not be used as an argument against the 
emplo^^nent of a virtual profile. The virtual profile shows the 
actual state of the case and only points out the necessity, if an 
unexpected requirement for a full stoppage of trains at a critical 
point has developed, of changing the location (if a station), or 
of changing the grade by regrading or by using an overhead 
crossing. Examples of such modifications are given in Chap- 
ter XXIV, The Improvement of Old Lines. 

434. Undulatory grades. Advantages. Money can generally 
be saved by adopting an actual profile which is not strictly 
uniform — the matter of compensation for curvature being here 



§ 434. GRADE. 515. 

ignored. Its effect on the operation of trains is harmless pro- 
vided the sag or hump is not too great. In Fig. 214 the undu- 
latory grade may actually be operated as a uniform grade AG. 
The sag at C must be considered as a sag, even though BC is actu- 
ally an up grade. But the engine is supposed to be working 




Fig. 214. 
hard enough to carry a train at uniform velocity up a grade AG. 
Therefore it gains in velocity from B to C, and from C to D loses 
an equal amount. It may even be proven that the time re- 
quired to pass the sag will be slightly less than the time required 
to run the uniform grade. 

Disadvantages. The hump at F is dangerous in that, if the 
velocity at E is not equal to that corresponding to the extra 
velocity-head ordinate at Fy the train will be stalled before 
reaching F, In practice there should be considerable margin. 
Any train should have a velocity of at least 10 miles per hour 
in passing any summit. This corresponds to a velocity head 
of 3.51 feet. An extra heavy head wind, slippery rails, etc., 
may use up any smaller margin and stall the train. If the 
grade AG is a ruling grade, then no hump should be allowed 
under any circumstances. For the heaviest trains are supposed 
to be so made up that the engine will just haul them up the 
ruling grades — of course with some margin for safety. Any 
mcrease of this grade, however short, would probably stall the 
train. 

Safe limits. It is quite possible to have a sag so deep that 
it is not safe to allow freight trains to rush through them with- 
out the use of brakes. The use of brakes of course adds a 
distinct element of cost. To illustrate: If a freight train is 
running at a velocity of 20 miles per hour (velocity head 14.05 
feet) and encounters a sag of 25 feet, the velocity head at the 
bottom of the sag will be 39.05 feet, which corresponds to a 
velocity of 33.3 miles per hour. This approaches the limit of 
safe speed for freight trains, and certainly passes the limit for 
trains not equipped with air-brakes and automatic couplers. 



516 RAILROAD CONSTRUCTION. § 434. 

The term ''safe limits " as used here, refers to the limits within 
which a freight train may be safely operated without the appli- 
cation of brakes or varying the work of the engine. Of course 
much greater undulations are frequently necessary and are 
safely operated, but it should be remembered that they add a 
distinct element to the cost of operating trains and that they 
must not be considered as harmless or that they should be 
introduced unless really necessary. 



MINOR GRADES. 

435. Basis of cost of minor grades. The basis of the com- 
putation of this least objectionable form of grade is as follows: 
The resistance encountered by a train on a level straight track 
is somewhat variable, depending on the velocity and the num- 
ber and character of the cars, but for average velocities we 
may consider that 10 lbs. per ton is a reasonable figure. This 
value agrees fairly well with the results of some dynamometer 
tests made by Mr. P. H. Dudley, using a passenger train of 
313 tons running at about 50 miles per hour. It also agrees 
with the Engineering News formula (Eq. 143) for the re- 
sistance of a train at a velocity of 32 miles per hour. Ten 
pounds per ton is the grade resistance of a 0.5% grade, or 
that of 26.4 feet per mile. On the above basis, a 0.5% grade 
will just double the tractive resistance on a level straight track. 
We may compute, as in the previous chapter, the cost of doubling 
the tractive resistance for one mile. But since the extra resist- 
ance is due to lifting the train through 26.4 feet of elevation, 
we may divide the extra cost of a mile of 0.5% grade by 26.4 
and we have the cost of one foot of difference of elevation, and 
then (disregarding the limiting effect of grades) we may say that 
this cost of one foot of difference of elevation will be independent 
of the rate of grade. There are, however, limitations to this 
general proposition which will be developed in the next section. 

436. Classification of minor grades. These are classified with 
reference to their effect on the operation of trains. In the first 
class are grades which may be operated without changing the 
work of the engine and which have practically no other effect 
than a harmless fluctuation of the velocity. But a grade which 
belongs to this class when considering a fast passenger train will 
belong to another class when considering a slow but heavy 




§ 436. GRADE. 517 

freight train. And since it is the slow heavy freight trains 
which must be chiefly considered^ a grade will usually be classi- 
fied with respect to them. The limit of class A (the harmless 

class) therefore depends on the 
^ maximum allowable speed. The 
effect of a sag on speed will depend 
"■J^^C^^^^^^^j/ on the vertical feet of drop rather 

than on the rate of grade, for 
with the engine working as usual 
on even a light down grade a train 
would soon exceed permissible speed. Assume that a freight 
train runs at an average speed of 15 miles per hour mth a 
minimum of 10 miles and a permissible maximum of 30 miles 
per hour. Assume that a train runs up the grade at A with 
a uniform velocity of 15 miles per hour, i.e., the engine is 
working so that the velocity would be uniform to C. How 
much sag {BB'^ can there be without the speed exceeding 30 
miles per hour? . 

Velocity head for 30 miles per hour 31 . 60 

'' '^ '' 15 '' '' '' 7.90 

The drop BB' will therefore be 23.70 

While each case must be figured by itself, considering the 
probable velocity of approach and the maximum permissible 
velocity, we may say that a sag of ahout 24 feet will ordinarily 
mark the limit of this class. With a higher velocity of approach 
even this limit will be much reduced. 

The classification therefore applies to sags and humps and 
to the vertical feet of drop or climb which are involved, rather 
than to grade per se. The practical application of these prin- 
ciples is necessarily confined to humps or sags which are pos- 
'sibly removable and does not apply to the long grades which 
are essential to connect predetermined points of the route — 
grades which are irreducible except by development and which 
must be studied as ruling grades (see §§ 440-445). 

The application therefore consists in the comparative study 
of two proposed grades, noting the relative energy required 
to operate them and the probable cost. The depth in feet 
saved would be the maximum difference between the grades, 
and the classification will depend on the necessary method of 
operating the trains. 



518 RAILROAD CONSTRUCTION. § 436. 

The next classification (B) applies to drops so deep that 
steam must be shut off when descending the grade, while the 
work required of the engine when ascending the opposite grade 
is correspondingly increased. The loss is not so serious as 
in the next case, but the inability of the engine to work con- 
tinuously may result in a failure to accumulate sufficient kinetic 
energy to carry the train over a succeeding summit. 

The third class (C) includes the grades so long that brakes 
must be applied to prevent excessive velocity. The loss in- 
volved is very heavy; the brakes require power for their appli- 
cation, they wear the brake-shoes and wheel-tires, they destroy 
kinetic or potential energy which had previously been created, 
while the tax on the locomotive on the corresponding ascend- 
ing grade is very great. The ascending grade may or may not 
be a ruling grade. 

437. Effect on operating expenses. As in the previous chapter 
we may at once throw out a large proportion of the items of 
expense of an average train-mile. In ''maintenance of way and 
structures/' Items 2 to 6 will be variously affected according 
to the classification of grades. The other items are evidently 
unaffected. 

Item 3. Renewals of ties. The extra wear of ties on 
grades is considered by Wellington as being somewhat com- 
pensated by the improved drainage of the roadbed which is 
found on a track which is not quite level, the better drainage 
tending to increase the life of the ties. On this account Welling- 
ton made an allowance of 5% increase for class C and no increase 
for the other classes. 

Item 4. Renewals of rails. Observations of rail wear on 
heavy grades show that the wear is much greater than on level 
tangents. The heavy grades of a mountain division of a road 
are usually operated with shorter trains or with the help of pusher- 
engines, and in such cases the proportion of engine tonnage to 
the total is much greater than on minor grades, and, since the 
engine has much greater effect on rail wear than an equal 
total tonnage of cars, partly on account of the use of sand 
and excess of engine tonnage, the grade will have a marked 
effect on rail wear. But such circumstances apply to ruling 
grades and very little, if at all, to minor grades. The use of 
sand on up-grades and the possible skidding of wheels on down- 
grades will wear the rails considerably. Even the sHpping 



§ 437 GRADEo 519 

of the drivers, although sand is not used, will wear the rails. 
Wellington allows 10% increase for class C and 5% for class B. 

Item 6. Roadway and track. It is very plain that a 
large proportion of the subitems are absolutely unaffected by 
minor grades. In fact it is a Httle difficult to ascribe any definite 
increase to any subitem. Rail wear is somewhat increased 
and this will have some effect in increasing the track- work. 
WeUington aUows 5% increase as a hberal estimate for class C, 
and makes no allowance for classes A and B. Items 2 and 5 
are allowed the same. 

Maintenance of equipment. It is evident that none of these 
items will be affected except repairs and renewals and depre- 
ciation of road engines and cars. The chief effect on these items 
will evidently be the repairs and renewals of wheels and brake- 
shoes. The draw-bars are also apt to suffer somewhat, on 
account of the increased work which they are required to do. 
It would appear that the extra stress on the locomotive mechan- 
ism wiU have some effect on locomotive repairs, and the expense 
of boiler repairs may be increased on account of the greater 
range of the demands on it. It would seem as if such effects 
would be quite large, but if the records of engine and car repairs 
on mountain divisions and on comparatively level divisions 
of the same road are examined and compared, there appears 
to be no such difference as might be expected. Considering the 
very small proportion of locomotive and car repairs which are 
affected at all by such circumstances, and also the very small 
percentage of these subitems which can be said to be affected by 
these two classes of minor grades, Wellington allows only an 
increase of 4% on each of these four items for class C and 1% 
for class B. This is reasonable in view of the 19% allowed 
for grades and curvature (see § 406) of which 13% was estimated 
for curves, leaving only 6% for all grades. 

Conducting transportation. It is apparent that the only 
items in conducting transportation which will be affected will 
be the four items of engine supplies (82 to 85). As in Chapter 
XXII, since we are considering that the resistance is doubled, 
we will assume that there is an increase of 44% as the cost 
of the extra fuel used in climbing 26.4 feet, but the total cost 
of both rise and fall is to be considered. In class B, although 
steam is shut off, fuel will be wasted by mere radiation. On 
this account we will add 5%. For class C we must allow in 



520 



RAILROAD CONSTRUCTION. 



§437. 



addition the energy spent in applying the brakes, which we 
may assume as 5% more. Therefore we will allow 49% for 
class B and 54% for class C. The other small items of suppUes, 
83, 84, and 85, will be estimated similarly. The items of general 
expense are evidently unaffected. 

438. Estimate of cost of one foot of change of elevation. Col- 
lecting the above estimates we have Table XXIV, showing 



TABLE XXIV. EFFECT ON OPERATING EXPENSES OF 26.4 FEET 

OF RISE AND FALL. 





Item (abbreviated). 


Normal 
average 


Class B. 


Class C. 


Item 
number? 


Per 

cent 

afifected 


Cost 

per mile 

per 

cent. 


Per 

cent 
affected. 


Cost 

per mile 

per 

cent. 


2 
3 
4. 
5 
6 


Ballast 

Ties 

Rails 

Other track material. . 
Roadway and track. . . 
(All other items) 

Maintenance of way. . . 


0.50 
3.10 
0.92 
1.13 
7.53 
6.91 




5 









0.05 








5 
5 
10 
5 
5 



0.02 
0.15 
0.09 
0.06 
0.38 





20.09 




0.05 




0.70 


25-27, 
31-36, 
43-45 


[Locomotives, passen- 
\ ger cars, freigh -cars 

[ and Ttork-cars 

(All othe items) 

Maintenance of equip- 
ment 


21 . '4 
1.50 


1 



0.21 



4 



0.85 





22.74 




0.21 




85 








53-60 


Traffic 


3.08 














82-85 


Fuel and other supplies 

for road engines 

(All other items) 

Transportation 


11.41 
39.03 


49 



5.59 



54 



6.16 





50.44 




5.59 




6.16 


106-116 


General expenses 


3.65 


















100.00 




5.85 




7.7M 



that the percentage of increase for operating grades of classes B 
or C will be 5.85% and 7.71% respectively on the average cost 
of a train-mile. On the basis of an average cost of $1.50 per 
train-mile, the additional cost for the 26.4 feet of rise and fall 
in one mile would be 8.77 c. and 11.56 c, or 0.33 c. and 0.44 c. 



439. 



GRADE. 



521 



per foot for the two classes respectively. For each train per day 
each way per year the value per foot of difference of elevation is 

For class J5: 2X365 X$0.0033 =$2.41. 
For class C: 2 X365X«0.0044 = $3.21.3 

In assuming the number of trains running over a given grade 
the character of the trains. must be carefully considered, since 
it will quite frequently happen that a hump or sag must be 
classified as belonging to class (7, so far as heavy freight-trains 
are concerned, but may be classified as B or perhaps even A, 
the harmless class, for passenger-trains. The above values 
should Hkewise be modified when the trains are run only 313 
days per year instead of 365. 

439. Operating value of the filling of a sagin a grade- Assume 
that the sag is 4000 feet long and that its depth in the center 
is 35 feet, as sketched in Fig. 216. Assume that a freight train 
is approaching the sag from the right-hand side (running to the 
left), the speed of passing D being 25 miles per hour. The 0.3% 
grade to the left will furnish a gravity pull of 6 pounds per ton, 
which may be more than half the force required to pull the train, 
and the locomotive would have but little to do, even if there 
were no sag, to maintain the speed of 25 miles per hour. Assum- 
ing that the locomotive is doing just this amount of work in 
running to the left through the sag, it will gain in velocity. 
Its velocity head at D is that corresponding to 25 miles per 
hour, or 21.95 feet. Adding 35 feet, the depth of the sag, 
we have 56,95 feet, which is the velocity head at C, which 




means that the velocity of the train would be 40.3 miles per 
hour. For passenger trains this will not be an objectionably 
high velocity, and even freight trains, which are provided with 
air-brakes and standard M. C. B. couplers, which are now 



522 RAILROAD CONSTRUCTION. § 439. 

nearly universal, may be safely run at this velocity. Therefore 
for all trains which may run at a speed of over 40 miles per 
hour, this sag will belong to class A, or the harmless class, so 
far as trains moving to the left are concerned. The effect on 
trains moving to the right will depend partly on the rate of 
ruling grade on that section of the road. The conditions of 
class A pre-suppose that the draw-bar pull is constant, but horse- 
power is measured by the product of pull times velocity. As- 
sume that AB is 3i ruling grade — 0.3% has recently been adopted 
as ruling grade for revision work on the Erie R. R. Then, if 
20 miles per hour were the speed of approach (running to the 
right) the speed at C would have to be 37.4 miles per hour. 
But since the draw-bar pull is to be constant, the horse-power 

/37.4\ 
developed would have to be nearly doubled ( — — ) . Since 



this would be impossible, it would mean that such a sag would 
not only be a serious matter but would be prohibitive on a 
ruling grade. For ordinary roads, 0.3% will not be a ruling 
grade, and the possibility of temporarily increasing the horse- 
power developed by the locomotive while running through the 
sag, so that the draw-bar pull will remain constant, will be far 
greater. The ability to develop such horse-power is very apt 
to be the criterion as to whether a sag belongs to class A rather 
than the danger that the speed may be prohibitive. The crite- 
rion as to whether the grade belongs to class B or C for trains 
moving to the left, depends on whether brakes must be applied 
before reaching the bottom of the sag. A sharp curve at or 
near C might require the use of brakes to prevent a dangerous 
velocity. For trains moving to the right, there is no definite 
criterion between classes B and (7, but a 2.05% grade is a very 
severe tax on a locomotive, especially when the assistance of 
momentum has been wasted by shutting off steam or by the 
applicatioii of brakes. Ignoring the possible limiting effect of 
a ruling grade (which is a separate matter) the value of the 35- 
foot sag is evidently 

35XS2.41 = $84.35 per daily round-trip train for class B 
and 

35X$3.21 =.$112.35 per daily round-trip train for class C. 

Assume that there are six daily trains each way, for which 
the grade would be classified as grade B, and four others, for 



§ 440. GRADE. 523 

which the sag would be classified as involving class C grade. 
Then on the above basis, the total annual cost would be 

6X $84.35=1506.10 
4X$112.35 =$449.40 



$955.50 



This annual cost capitalized at 5% equals $19,110, which is 
the justifiable expenditure to fill up the sag. The amount of 
fill in such a sag, roughly calculated, is 125,000 cubic yards. 
Assuming that it would cost 30 c. per cubic yard to make the 
fill, this would require an expenditure of $37,500, which appar- 
ently would not be justifiable. 

But another solution may be considered. It has been shown 
above that a sag may be made harmless for all classes of trains 
provided the depth is not greater than some uncertain limit, 
which depends on the particular circumstances of the case. 
The volume of earth in this fill is great on account of the great 
<lepth in the center. It may be readily computed that by fill- 
ing up the lower part of the sag so that its maximum depth 
below the grade line, BED, is about one-half of CE, or about 
17 feet, that the amount of earthwork required will be only 
about 25,000 yards rather than 125,000 yards. This would 
make the cost of such a fill practically $7500 rather than $37,500. 
If it could be shown that a sag of about 18 feet could be operated 
on the class A basis by all trains, then it would certainly be 
justifiable to expend $7500 in order to secure a reduction in the 
operating expenses, whose capitalized value according to the 
above calculation, is $19,110. 

RULING GRADES. 

440. Definition. Ruling grades are those which limit the 
weight of the train of cars which may be hauled by one engine. 
The subject of ''pusher grades'' will be considered later. For 
the present it will suffice to say that on all well-designed roads 
the large majority of the grades on any one division are kept 
below some limit which is considered the ruling grade. If a 
heavier grade is absolutely necessary no special expense will 
be made to keep it below a rate where the resistance is twice 



624 RAILROAD CONSTRUCTION. § 440, 

(or possibly three times) the resistance on the ruHng grade, and 
then the trains can be hauled unbroken up these few special 
grades with the help of one (or two) pusher engines. So far 
as limitation of train length is concerned, these pusher grades 
are no worse than the regular ruling grades and, except for the 
expense of operating the pusher engines (which is a separate 
matter), they are not appreciably more expensive than any 
ruling grade. As before stated, the engineer cannot alter very 
greatly the ruling grade of the road when the general route has 
been decided on. He may remove sags or humps, or he may 
lower the natural grade of the route by development in order 
to bring the grade within the adopted limit of ruling grade. 
The financial value of removing sags and humps has been con- 
sidered. It now remains to determine the financial relation 
between the lowest permissible ruling grade and the mor^y 
which may profitably be spent to secure it. 

441. Choice of ruling grade. It is of course impracticable for 
an engine to drop off or pick up cars according to the grades 
which may be encountered along the line. A train load is made 
up at one terminus of a division and must run to the other 
terminus. Excluding from consideration any short but steep 
grades which may always be operated by momentum, and also 
all pusher grades, the maximum grade on that division is the 
ruling grade. 

It will evidently be economy to reduce the few grades which 
naturally would be a little higher than the great majority of 
others until such a large amount of grade is at some uniform 
limit that a reduction at all these places would cost more than 
it is worth. The precise determination of this limit is prac- 
tically impossible, but an approximate value may be at once 
determined from a general survey of the route. The distance 
apart of the termini of the division into their difference of ele- 
vation is a first trial figure for the rate of the grade. If a grade 
even approximately uniform is impossible owing to the eleva- 
tions of predetermined intermediate points, the worst place 
may be selected and the natural grade of that part of the route 
determined. If this grade is much steeper than the general 
run of the natural grades, it may be policy to reduce it by devel- 
opment or to boldly plan to operate that place as a pusher 
grade. The choice of possible grades thus has large limita- 
tions, and it justifies very close study to determine the best 



§442. 



GKADE. 



525 



combination of grades and pusher grades. When the choice 
has narrowed down to two limits, the lower of which may be 
obtained by the expenditure of a definite extra sum, the choice 
may be readily computed, as will be developed. 

442. Maximum train load on any grade. The tractive power 
of a locomotive has been discussed in Chap. XV, § 322. The 
net train load w^hich may be placed Behind any engine is the 
difference between the weight of the engine itself and the gross 
load which can be handled under the given circumstances, wdth 
a given weight on the drivers. Since the design of locomotives 
is so variable, it is impracticable to show in tabular form the 
power of all kinds of locomotives on all grades. In Table XXV 
are given the tractive powers of locomotives of a wide range of 
types and weights and with various ratios of adhesion. They 
may be accepted as typical figures and wdll serve to compute the 
effect of variations of grade on train load. In Table XXVI is 
given the total train resistance in pounds per ton for various grades 
and for various values of track resistances. By a combination 

TABLE XXV. TRACTIVE POWER OF VARIOUS TYPES OF STANDARD- 
GAUGE LOCOMOTIVES AT VARIOUS RATES OF ADHESION. 



Type of locomotive. 


Total weight 

of engine 
and tender. 


Weight 

of 

engine 

only. 


Weight 

on the 
drivers. 


Tractive power when 

ratio of adhesion 

is 




Lbs. 


Tons. 


i 


^% 


\ 


Atlantic, 4-4-2 .... 

Atlantic, 4-4-2, four 

cylinder compound 

Pacific, 4-6-2 

Pacific, 4-6-2 

Ten-wheel, 4-6-0 . . . 

Prairie, 2-6-2 

Consolidation, 2-8-0 
Consolidation, 2-8-0 
Mikado, 2-8-2 


340,000 

368,800 
343,600 
403,780 
321,000 
366,500 
214,000 
366,700 
405,500 


170.0 

184.4 
171.8 
201.9 
160.5 
183.2 
107.0 
183.3 
202.7 


199,400 

206,000 
218,000 
226,700 
201.000 
212,500 
120,000 
221,500 
259,000 


105,540 

115,000 
142,000 
151,900 
154,000 
154,000 
106,000 
197,500 
196,000 


26,385 

28,750 
35,500 
37,975 
38,500 
38,500 
26,500 
49,375 
40.000 


23,740 

25,875 
31,950 
34,180 
34,650 
34,650 
23,850 
44,440 
44,100 


21,100 

23,000 
28,400 
30,380 
30,800 
30,800 
21,200 
39,500 
39,200 



of these two tables the net train load on any grade under given 
conditions may be quickly determined For example, an 
ordinary consolidation engine having a weight of 106000 
pounds on the drivers (see Table XXV) will have a tractive 
force of 26500 pounds under fair conditions of track, when the 
adhesion ratio is \. When climbing slowly up a grade of 1.30% 
the tractive resistance will be about 32 pounds per ton if the roll- 
ing-stock and track are fair — assuming a tractive resistance on 



526 RAILROAD CONSTRUCTION. § 442. 

a level of 6 pounds per ton. Dividing 26500 by 32 we have 
828 tons, the gross train load. Subtracting 107 tons, the weight 
of the engine and tender in working order, we have 721 tons, 
the net load. Incidentally we may note that, cutting down 
the grade to 0.90% (a reduction of only 21.12 feet per mile), 
the resistance per ton is reduced to 24 pounds and the gross 
train load is increased to 1104 tons and the net load to 997 
tons — a.n increase of about 38%. 

As another numerical example, consider a contractor's loco- 
motive (not referred to in Table XXV), a light four-wheel-con- 
nected-tank narrow-gauge engine, with a total weight of 12000 
pounds, all on the drivers. On the rough temporary track 
used by contractors the tractive ratio may be as low as ^. 
The tractive adhesion should therefore be taken as 2400 pounds. 
Assume that the grade when hauling "empties'' is 4.7% and 
that the tractive resistance on such a track on a level is 10 pounds 
per ton. By Table XXVI, the total train resistance is therefore 
(by interpolation) 104 pounds per ton. 2400 -^ 104 = 23 tons; 
subtracting the weight of the engine we have 17 tons, the net 
load of empty cars — ^perhaps twenty cars weighing 1700 pounds 
per car. 

In general,' and to compute accurately the train load under 
conditions not exactly given in the tables, the maximum train 
load at slow speed may be computed according to the following 
rule: 

The maximum load behind an engine on any grade may be 
found by multiplying the weight on the drivers by the ratio of 
adhesion and dividing this by the sum of the grade and tractive 
resistances per ton; this gives the gross load, from which the 
weight of the engine and tender must be subtracted to find the 
net load. 

443. Proportion of the traffic affected by the ruling grade. 
Some very light traffic roads are not so fortunate as to have 
a traffic which will be largely affected by the rate of the ruling 
grade. When passenger traffic is light, and when, for the sake 
of encouraging traffic, more frequent trains are run than are 
required from the standpoint of engine capacity, it may happen 
that no passenger trains are really limited by any grade on the 
road — i.e., an extra passenger car could be added if needed. 
The maximum grade then has no worse effect (for passenger 
trains) than to cause a harmless reduction of speed at a few points. 



§443. 



GRADE. 



527 



TABLE XXVI. TOTAL TRAIN RESISTANCE PER TON (OF 2000 

pounds) on YAEICU3 GRADES. 







When tractive 


re- 


Grade. 


sistance on a level 






in pounds per ton is 


Rate 


Feet 












per 


per 


6 


/ 


8 


9 


10 


cent. 


mil3. 












0.00 


0.00 


6 


7 


8 


9 


10 


.05 


2.61 


7 


8 


9 


10 


11 


.10 


5.28 


8 


9 


10 


11 


12 


.15 


7.92 


9 


10 


11 


12 


13 


.20 


10.56 


10 


11 


12 


13 


14 


0.25 


13.20 


11 


12 


13 


14 


15 


.30 


15.84 


12 


13 


14 


15 


16 


.35 


18.48 


13 


14 


15 


16 


17 


.40 


21.12 


14 


15 


16 


17 


18 


.45 


23.76 


15 


16 


17 


18 


19 


0.50 


26.40 


16 


17 


18 


19 


20 


.55 


29.04 


17 


18 


19 


20 


21 


.60 


31.68 


IS 


19 


20 


21 


22 


.65 


34.32 


19 


20 


21 


22 


23 


.70 


36.96 


20 


21 


22 


23 


24 


0.75 


39.60 


21 


22 


23 


24 
25 


25 
26 


.80 


42.24 


22 


23 


24 


.85 


44.88 


23 


24 


25 


26 


27 


.90 


47.52 


24 


25 


26 


27 


28 


0.95 


50.16 


25 


26 


27 


28 


29 


1.00 


52.80 


26 


27 


28 


29 


30 


.05 


55.44 


27 


28 


29 


30 


31 


.10 


58.08 


28 


29 


30 


31 


32 


.15 


60.72 


29 


30 


31 


32 


33 


.20 


63.36 


30 


31 


32 


33 


34 


1.25 


63.00 


31 


32 


33 


34 


35 


.30 


68.64 


32 


33 


34 


35 


36 


.35 


71.28 


33 


34 


35 


36 


37 


.40 


73.92 


34 


35 


36 


37 


38 


.45 


76.56 


35 


36 


37 


38 


39 


1.50 


79.20 


36 


37 


38 


39 


40 


.55 


81.84 


37 


38 


39 


40 


41 


.60 


84.48 


38 


39 


40 


41 


42 


.65 


87.12 


39 


40 


41 


42 


43 


.70 


89.76 


40 


41 


42 


43 


44 


1.75 


92.40 


41 


42 


43 


44 


45 


.80 


95.04 


J.o 


43 


44 


45 


46 


.85 


97.68 


43 


44 


45 


46 


47 


.90 


100.32 


44 


45 


46 


47 


48 


1.95 


102.96 


45 


46 


47 


48 


40 


2.00 


105.60 


46 


47 


48 


49 


50 







When tractive 


re- 


Grade. 


sistance on 


a level 






in pounds per ton is 


Rate 


Feet 












per 


per 


6 


7 


8 


9 


10 


cent. 


mile. 












2.00 


105.60 


46 


47 


48 


49 


50 


.05 


108.24 


47 


48 


49 


50 


51 


.10 


110.88 


48 


49 


50 


51 


52 


.15 


113.52 


49 


50 


51 


52 


53 


.20 


116.16 


50 


51 


52 


53 


54 


2.25 


118.80 


51 


52 


53 


54 


55 


.30 


121.44 


52 


53 


54 


55 


56 


.35 


124.08 


53 


54 


55 


56 


57 


.40 


126.72 


54 


55 


56 


57 


58 


.45 


129.36 


55 


56 


57 


58 


59 


2.50 


132.00 


56 


57 


• 58 


59 


60 


.55 


134.64 


57 


58 


59 


60 


61 


.60 


137.28 


58 


59 


60 


61 


62 


.65 


139.92 


59 


60 


61 


62 


63 


.70 


142.56 


60 


61 


62 


63 


64 


2.75 


145.20 


61 


62 


63 


64 


65 


.80 


'147.84 


62 


63 


64 


65 


66 


.85 


150.48 


63 


64 


65 


66 


67 


.90 


153.12 


64 


65 


66 


67 


68 


.95 


155.76 


65 


66 


67 


68 


69 


3.00 


158.40 


66 


67 


68 


69 


70 


.05 


161.04 


67 


68 


69 


7C 


71 


.10 


163.68 


68 


69 


70 


71 


72 


.15 


166.32 


69 


70 


71 


72 


73 


.20 


168.96 


70 


71 


72 


73 


74 


3.25 


171.60 


71 


72 


73 


74 


75 


.30 


174.24 


72 


73 


74 


75 


76 


.35 


176.88 


73 


74 


75 


76 


77 


.40 


179.52 


74 


75 


76 


77 


78 


.45 


182.16 


75 


76 


■ 77 


78 


79 


3.50 


184.80 


76 


77 


78 


79 


80 


4.00 


211.20 


86 


87 


88 


89 


90 


4.50 


237.60 


96 


97 


98 


99 


100 


5.00 


264.00 


106 


107 


108 


109 


110 


5.50 


290.40 


116 


117 


118 


119 


120 


6.00 


316.80 


126 


127 


128 


129 


130 


6.50 


343 . 20 


136 


137 


138 


139 


140 


7.00 


369.60 


146 


147 


148 


149 


150 


8.00 


422.40 


166 


167 


168 


169 


170 


9.00 


475.20 


186 


187 


188 


189 


190 


10.00 


528 . 00 


206 


207 


208 


209 


210 



The local freight business is frequently affected in practically 
the same way. All coal, mineral, or timber roads are affected 



528 KAILROAD CONSTRUCTION. § 443, 

by the rate of ruling grade as far as such traffic is concerned. 
Likewise the through business in general merchandise, especially 
of the heavy traffic roads, will generally be affected by the rate 
of ruling grade. Therefore in computing the effect of ruling 
grade, the total number of trains on the road should not ordi- 
narily be considered, but only the trains to which cars are added, 
until the limit of the hauling power of the engine on the ruling 
grades is reached. 

444. Financial value of increasing the train load. The 
gross receipts for transporting a given amount of freight is a 
definite sum regardless of the number of train loads. The 
cost of a train mile is nearly constant. If it were exactly so, the 
saving in operating expenses would be strictly proportional 
to the number of trains saved. How will the cost per train 
mile vary when by a reduction in ruling grade more cars are 
handled in one train than before? First, compute the effect 
of increasing the train load so that one less engine will handle 
the traffic, or, for example, that an engine can haul 11 cars 
instead of 10 or 44 instead of 40 — that 10 engines will do the 
work for which 11 engines would be required with the steeper 
grade. What will be the relative cost of running 10 heavy 
trains rather than 11 Hghter trains, or, rather, what will be the 
extra cost of the extra engine? 

We will estimate as before the difference in the cost of operating, 
say, four light trains on heavy grades, or three heavier trains 
on the Hghter grades. In either case the gross tonnage of cars, 
with their contents, is supposed to be the same. The difference 
consists in the cost of operating the extra engine and also the 
extra cost for train service, etc., which is a function of the 
number of trains on the road rather than of their tonnage. The 
additional cost of maintenance of way is confined to the effect 
of the extra engine, and this will evidently effect only Items 2 
to 6 and 9. On the basis that an engine produces one-half of 
the track deterioration, we may allow 50% of those items as 
the total effect on maintenance of way. 

Maintenance of equipment. The effect on maintenance of 
equipment will be practically confined to the repairs, renewals, 
and depreciation of steam locomotives (Items 25 to 27) and the 
same items for freight-cars (Items 34 to 36). Very few roads 
have a passenger traffic which is affected by the rate of the 
ruHng grade, since, for the encouragement of traffic, passenger- 



§ 444. GRADE. 529 

trains are usually added to the schedule in advance of the 
physical capacity of a locomotive to haul one or more addi- 
tional cars. Therefore, in general, no allowance need be made 
for any effect on passenger-trains, or on the maintenance of the 
passenger-cars. But the cost of maintaining the freight-cars is 
actually reduced by having more trains and less cars per train. 
This means that, although the maximum draw-bar pull will be the 
same in both cases, and will equal the maximum capacity of the 
locomotive, while on the ruhng grades, the draw-bar pull while 
on the Hght grades, level track and down grades, which may 
mean 90% of the length of the road, will average very much 
less. It is impossible to make an accurate estimate of the 
amount of this saving which would be generally apphcable. 
WelHngton estimated it at 10%. Considering the very|^large 
proportion of freight-car maintenance charges which are evi- 
dently independent of draw-bar pull, the estimate is probably 
large enough and the error in adopting that figure is not very 
great. ^ 

Although, for either system of grades, the locomotive is sup- 
posed to work to its full capacity while on the ruHng grades, 
there is also some saving for each locomotive when hauhng a 
Hghter train over the Hght grades, level sections and down 
grades. If, on account of the reduction in average draw-bar 
pull, the repairs of each of four locomotives were reduced 5% 
below the repairs of the three locomotives which could haul 
the same number of cars over lower ruhng grades, then the re- 
pairs of the four locomotives would cost 4X95, or 380 com- 
pared with 3X100, or 300 for the three locomotives. This 
would mean that the additional locomotive should be assigned 
an added expenditure of 80% of the average cost for one. If 
the saving by the reduction of grade was only half as much, 
or one train in eight, so that seven trains were required to do 
the work of eight on the steeper grade, then the saving per 
engine would be correspondingly less. If it were 2.5% instead 
of 5, we would have, for the cost of eight engines, 8X97.5, or 
780 compared with 7 X 100, or 700 for the seven engines. Again, 
we would have 80% as the additional net cost of the repairs on 
the additional engine. Of course the above estimates of 5% 
and 2.5%, as the saving on one engine, are merely guesswork, 
but the above demonstration shows that if the saving in repairs 
Is proportional to the reduction in number of trains which is 



530 RAILROAD CONSTRUCTION. §444. 

made possible by the reduction of grade, as is quite probable, 
then the cost of repairs of the extra engine is the same, whether 
it is one engine out of four, or one engine but of eight, or one 
engine out of any other number. Therefore, although the 
estimate of 5% per engine as made above is a guess, it is prob- 
ably near enough to the truth, so that there is a comparatively 
httle error in using the figure of 80% for the additional cost of 
repairs of the additional engine. 

Conducting transportation. Items 61-79, chiefly yard work, 
will be practically unaffected. Items 80 and 81 will be given 
their full value. The additional cost for the fuel for the added 
engine may be computed somewhat on the same basis as the 
cost of engine repairs. The fuel used by four engines will 
average somewhat less than that used by three engines, since 
the four engines will do far less work on the level and on very 
light grades, while on the heavier grades the engines are work- 
ing to the Hmit of their capacity in either case. The loss of 
heat, due to radiation and the other causes, which are inde- 
pendent of the direct work done by the engine in hauling, will 
be the same in either case. These causes have already been 
discussed in §407. It is impossible to make any general 
calculations as to the relative consumption of fuel in the two 
cases, since so much depends on the proportion of track which 
is level or which has a very light grade. If the four engines oper- 
ating lighter trains each burn 5% less fuel than three engines 
operating heavier trains, we will find, by the same method as 
before, that the extra engine may be charged with 80% of the 
average fuel consumption of the other three engines. If, as 
before, we assume that this variation in fuel consumption is 
proportional to the variation in the number of trains required 
to handle a given traffic, then the extra engine would be respon- 
sible for 80% of the average fuel consumption, regardless of 
whether the number of trains saved was one in four or one in 
ten. Although it is true that the value 80% is a mere guess, 
that no general value is obtainable, and that the value for any 
particular and special case could only be computed with great 
difficulty, it is evident that the error is not very great, and we 
will therefore assume 80% as the fuel consumption assignable 
to the extra engine. The consumption of other engine supplies, 
water, oil, waste, etc., is not strictly proportional to the con- 



§ 445. GRADE, 531 

sumption of fuel, but we will assume it to be so in this case, 
and that 80% of Items 83-85 are allowable for the extra 
engine. 

Items 88 to 94, which concern train-service, will be considered 
as varying according to the number of train-miles, and we wiU 
therefore* add 100% for all these items. Items 97 and 98 will 
be allowed 50%. Items 99, 101 to 103, which refer to damages, 
might be considered from one standpoint to be unaffected, 
while from other standpoints the effect might be considered as 
100%. The risk of train operations varies very largely with 
the number of trains, and yet in some respects the danger is 
independent of whether there are 15 or 20 cars in a train. There 
will be little error in assigning 50% extra for this item. Items , 
104 and 105 will be allowed 100% of their net value. The 
general expenses are evidently unaffected. Collecting these 
various items we have Table XXVII. 

If we assume that the average cost of a train-mile is $1.50, 
fchen the operating value per mile of saving the use of the addi- 
tional engine equals 41.63% of $1.50, or 62.45 c. 

445. Operating value of a reduction in the rate of the ruling 
grade. As a practical appHcation of the above figures, assume 
that on a constructed and operated road the ruling grade on a 
100-mile division is 1.6%; the actual traffic affected by ruHng 
grade is 8 daily trains with a net load of 552 tons or 4416 tons. 
It is found that with an expenditure of $400,000 the ruling grade 
maybe reduced to 1.2%. Will it pay? At 1.2% grade the net 
load behind an 80-ton consofidation engine, with 48 tons on 
the drivers, adhesion j, and 6 pounds per ton normal resistance, 
is 720 tons. The traffic (4416 tons) may therefore be hauled 
by 6 engines, the balance, less than 100 tons, being taken care 
of by fighter trains not affected by the ruling grade. There is 
therefore the saving due to not operating two engines. Since 
the additional cost of the two engines drawing fighter trains is 
62.45 c. per mile, the annual saving is therefore 2 X $0.6245X100 
X365 =$45,588.50, which capitafized at 5% =$911,770. This 
shows that if the improvement can be accomplished for $400,000 
it is worth while. 

As in other similar problems, it must be reiterated that al- 
though there are some more or less uncertain elements in the 
above estimates, yet with a considerable margin for error in 



532 



RAILROAD CONSTRUCTION. 



§445. 



TABLE XXVII. ADDITIONAL COST OF OPERATING A GIVEN FREIGHT 
TONNAGE WITH (n + 1) ENGINES ON HEAVY RULING GRADES 
INSTEAD OF WITH U ENGINES ON LIGHTER GRADES. 



Item 
number. 

1 


Item (abbreviated). 


Normal 
average. 


Per cent 
affected. 


Cost per 

mile, 
per cent. 


2 
3 
4 
5 
6 
9 


Ballast, 

Ties, 

Rails, 

Other track material, 

Roadway and track. 

Bridges, trestles and culverts 

(All other items) 




14.89 
5.20 


50 



7.45 





Maintenance of way 






20.09 




7.45 










25-27 


Steam locomotives 


8.61 

10.15 

3.98 


80 

-10 




6.89 


34-36 


Freight cars 


-1.01 




(All other items) 







Maintenance of equipment 






22.74 




5.88 










53-60 


Traffic 


3.08 














80 


Road enginemen 


6.08 
1.72 

11.41 
9.66 
0.57 

2.82 

0.02 

18.16 


100 
100 

80 

100 

50 

50 

100 




6.08 


81 
82-85 


Enginehouse expenses, road .... 

Fuel, and other supplies for road 

engines 


1.72 
9.13 


88-94 


Train service, etc 


9.66 


97, 98 


Stationery, printing, etc 


0.28 


99, 101, 
103 


Loss, damage, etc 


1.41 


104, 105 


Operating joint tracks, net 

(AH other items) 


0.02 





Transportation ,...,,.. 






50.44 




28.30 








106-116 


General expenses 


3.65 












100.00 




41.63 













individual items the value of the whole improvement will not 
be very greatly altered and the estimate will be infinitely better 
than an indefinite reliance on vague "judgment.'' Of course 
certain items in the above estimates are somewhat variable 
and should be altered to fit the particular case to be computed. 



PUSHER GRADES. 



446. General principles underlying the use of pusher engines. 
On nearly all roads there are some grades which are greatly 
in excess of the general average rate of grade and these heavy 



§446. GRADE. 533 

grades cannot usually be materially reduced without an ex- 
penditure which is excessive and beyond the financial capacity 
of the road. If no pusher engines are used, the length of all 
heavy trains is limited by these grades. The financial value 
of the reduction of such ruling grades has already been shown. 
But in the operation of pusher grades there is incurred the 
additional cost of pusher-engine service, for a pusher engine 
must run twice over the grade for each train which is assisted. 
It is possible for this additional expense to equal or even exceed 
the advantage to be gained. In any case it means the adoption 
of the lesser of two evils, or the adoption of the more economical 
method. A simple example wall illustrate the point. Assume 
that at one point on the road there is a grade of 1.9% which 
is five miles long. Assume that all other grades are less than 
0.92%. li pushers are not to be used the net capacit}^ of a 
107-ton consolidation engine with 53 tons on the drivers, assum- 
ing :f-Q adhesion and 6 pounds per ton for normal resistance, 
will be 435 tons, and that will be the maximum weight of train 
allowable. By using pusher engines on this one 5-mile grade 
the train load is at once doubled and the number of trains 
cut down one half. This double load, 870 tons, can easily be 
hauled by one engine up the 092% grades. As a rough com- 
parison, free from details and allowances, we may say: 

(a) 10 trains per day over a 100-mile division, 435 tons 
net per train, will require 1000 engine miles daily. 

(b) 5 trains per day handling the same traffic, 870 tons 
net per train, with 2X5X5 pusher-engine miles, will require 
(5X100) + (2X5X5) =550 engine miles daily. There is thus 
a large saving in the number of engine miles and also in the 
number of the engines required for the work. Moreover, the 
engines are working to the limit of their capacity for a much 
larger proportion of the time, and their work is therefore mxOre 
economically done. The work of overcoming the normal 
resistances of so many loaded cars over so many miles of track 
and of lifting so many tons up the gross differences of elevation 
of predetermined points of the line is approximately the same 
w^hatever the exact route, and if the grades are sc made that 
fewer engines w^orking more constantly can accomplish the 
work as well as more engines which are not hard worked for a 
considerable proportion of the time, the econom}^ is very ap- 



534 EAILEOAD CONSTRUCTION. § 446. 

parent and unquestionable. Wellington expresses it concisely: 
"It is a truth of the first importance that the objection to 
high gradients is not the work which the engines have to do 
on them, but it is the work which they do not do when they 
thunder over the track with a light train behind them, from 
end to end of a division, in order that the needed power may 
be at hand at a few scattered points where alone it is needed/' 

447. Balance of grades for pusher service. In the above 
illustration the "through'^ grade and the "pusher" grade are 
"balanced" for the use of one equal pusher. It is therefore 
evident that if some intermediate grade (such as 1.4%) were 
permitted, it could only be operated by (a) making it the ruling 
grade and cutting down all train loads from 870 tons to 594 tons, 
or (6) operating it as a pusher grade, although with a loss of 
economy, since two engines would have much more power thani 
necessary. The proper plan i*n such a case would be to strive 
to reduce the 1.4% grade to 0.92%, or, if that seemed imprac- 
ticable, to attempt to get an operating advantage at the expense 
of an increase of the 1.4% grade to anything short of 1.9%. 
For the increase in rate of grade would cost almost nothing, and 
some advantage might be obtained which would practically 
compensate for the introduction of a pusher grade. Another 
possible solution would be to operate the 1.9% with two pushers, 
adopt a corresponding grade for use with one pusher and a 
corresponding ruling grade for through trains. With the above 
data these three grades would be 1.90% 1.27%, and 0.54%, 
obtained as follows: 

Tractive power of three engines = 106000 X ^^ X 3 = 71550 
pounds. 

Resistance on 1.9% grade= 6+ (20X1.9) =44 lbs. per ton. 

71550-7-44= 1626 = gross load in tons. 

1626 - (3 X 107) = 1305 = net load in tons. ♦ 

1305 + (2X107) =1519=gross load on the one-pusher grade. 

Tractive power of two engines = 47700 lbs. 

47700 -^1519 =31.40= possible tractive force in lbs. per ton. 

(31.40 — 6)4-20= 1.27% = permissible grade for one pusher. 

1305 + 107= 1412 = gross load on the through grade. 

Tractive power of one engine = 23850 lbs. 

23850-^1412 =16.89 = possible tractive force in lbs. per ton. 

(16.89-6) -^ 20 =0.54% = permissible through grade. 



§447. 



GRADE. 



635 



It should be realized that, assuming the accuracy of the 
normal resistance (6 lbs.) and the normal adhesion (^^) and 
wdth the use of 107-ton locomotives with 53 tons on the drivers, 
the above figures are precisely what is required for hauling 
with one, two, and three engines. Other types of engines, other 
values for resistance and adhesion wdll vary considerably the 
gross load in tons which may be hauled up those grades, but 
starting with 0.54% as a through grade, the corresponding 
values for one and for two pushers would vary but slightly 
from those given. To show the tendency of these variations, 
the corresponding values have been computed as follows: 



Adhesion. 


Resistance 
per ton. 


Load on 
drivers. 


Through 
grade. 


One -pusher 
grade. 


Two-pusher 
grade. 


t 


6 lbs. 

7 " 
6 " 

6 " 

7 " 


53 tons. 
53 " 
53 " 
53 " 
53 " 


0.54% 
.54% 
.54% 
.54% 
.54% 


1.27% 
1.31% 
1.28% 
1.26% 
1.29% 


1.90% 
1.96% 
1.93% 
1.86% 
1.92% 



The above form shows that increasing the resistance per ton 
and decreasing the adhesion have opposite effects on altering 
the. ratio of these grades, and as a storm, for example, would 
increase the resistance and decrease the adhesion, the changes 
in the ratio would be eom.pensating although the absolute 
reduction in train load might be considerable. 

In Table XXVIII is shown a series of ^'balanced" grades on 
which a given net train load may be operated by means of one 
or two pusher engines. For example, assuming a track resistance 
of 6 pounds per ton, a consolidation engine of the type shown 
in the table can haul a train weighing 977 tons (exclusive of 
the engine) up a grade of 0.80%. If this is the maximum 
through grade, pusher grades as high as 1.70% for one pusher, 
or 2.46% for two pushers, may be introduced and the same 
net load may be hauled up these grades. 

The ratios of pusher grade to through grade, as given in 
Table XXVIII, are exactly true only for the conditions named 
as to weight and type of engine, ratio of adhesion, and norma 
track resistance. But a little comparative study of the two 



536 



RAILROAD CONSTRUCTION. 



§448. 



TABLE XXVIII. — BALANCED GRADES FOR ONE, TWO, AND 
THREE ENGINES. 
Basis. — Through and pusher engines alike; consolidation type; total 
weight, 107 tons; weight on drivers, 53 tons; adhesion, 3%, giving a trac- 
tive force for each engine of 23850 lbs.; normal track resistance, 6 (also 8) 
lbs. per ton. 





Track resistance, 


6 lbs. 


Track resistance, 


8 lbs. 






Corresponding 




Corresponding 


Through 


Net load 


pusher grade for 


Net load 


pusher grade for 


grade. 


for one 


same net load. 


for one 


same net load. 




engine in 
tons (2000 






engine in 
tons (2000 


















lbs.). 


One 


Two 


lbs.). 


One 


Two 






pusher. 


pushers. 




pusher. 


pushers. 


Level, 


3868 tons 


0.28% 


0.55% 


2874 tons 


0.37% 


0.72% 
0.98% 


0.10% 


2874 ' * 


0.47% 


0.82% 


2278 '* 


0.56% 


0.20% 


2278 *• 


0.66% 


1.08% 


1880 •' 


0.74% 


1.23% 


0.30% 


1880 " 


0.84% 


1.33% 


1596 " 


0.92% 


1.47% 


0.40% 


1596 •* 


1.02% 


1.57% 


1384 " 


1.09% 


1.70% 


0.50% 


1384 ♦' 


1.19% 


1.80% 


1218 " 


1.27% 


1.92% 


0.60% 


1218 " 


1.37% 


2.02% 


1085 " 


1.44% 


2.14% 


0.70% 


1085 " 


1.54% 


2.24% 


977 " 


1.60% 

1.77% 


2.36% 


0.80% 


977 " 


1.70% 


2.46% 


887 •• 


2.56% 


0.90% 


887 ♦' 


1.87% 


2.66% 


810 •' 


1.93% 


2.76% 


1.00% 


810 •' 


2.03% 


2.86% 


745 ** 


2.09% 


2.96% 
3.15% 


1.10% 


745 ** 


2.19% 


3.06% 


688 " 


2.24% 


1.20% 


688 •* 


2.34% 


3.25% 


638 ** 


2.40% 


3.33% 
3.51% 
3.68% 


1.30% 
1.40% 


638 •* 


2.50% 


3.43% 


594 *• 


2.55% 


594 •* 


2.65% 


3.61% 


555 •* 


2.70% 


1.50% 


555 " 


2.80% 


3.78% 


521 ♦• 


2.85% 


3.85% 
4.02% 


1.60% 


521 •* 


2.95% 


3.95% 


489 •• 


2.99% 


1.70% 


489 *• 


3.09% 


4.12% 


461 " 


3.13% 


4.17% 


1.80% 


461 •• 


3.23% 


4.27% 


435 •' 


3.27% 


4.33% 


1.90% 


435 •• 


3.37% 


4.43% 


411 ♦• 


3.42% 


4.49% 


2.00% 


411 ♦• 


3.52% 


4.59% 


390 •• 


3.55% 


4.63% 
4.78% 


2.10% 


390 " 


3.65% 


4.73% 


370 •• 


3.68% 


2.20% 


370 *• 


3.78% 


4.88% 


352 •• 


3.81% 
3.94% 


4.92% 


2.30% 


352 *• 


3.91% 


5.02% 


335 •• 


5.05% 


2.40% 
2.50% 


335 ♦' 


4.04% 


5.15% 


319 " 


4.07% 


5.19% 
5.32% 


319 •* 


4.17% 


5.29% 


304 •' 


4.20% 



halves of Table XXVIII and of the tabular form given on page 
535 will show that although the net load which can be hauled 
on any grade varies considerably with the normal track re- 
sistance and also with the ratio of adhesion, yet the ratios of 
through tio pusher grade, for either one or two pushers, varies 
but slightly with ordinary changes in these conditions. There- 
fore when the precise conditions are unknown or variable, the 



S^448. GRADBo ~ ^ 537 

figures of Table XXVIII may be considered as applicable to 
any ordinary practice, especially for preliminary computations. 
For final calculations on any proposed ruling grade and pusher 
grade, the whole problem should be worked out on the principles 
outlined above and on the basis of the best data obtainable. 

Problem : If the through ruHng grade for the road has been 
established at 1.12%, what pusher grades are permissible? 
Answer: Interpolating in Table XXVIII, we may employ a 
grade of 2.22% if the track and road-bed are to be such that a 
tractive resistance of 6 pounds per ton can be expected. With 
a poorer track, the normal resistance assumed as 8 pounds per 
ton, the rate is raised to 2.27%. The increase in rate of pusher 
grade with increase of resistance is due to the fact that the 
net load hauled is less — so much less that on the pusher grade 
a larger part of the adhesion is available to overcome a grade 
resistance. 

448. Operation of pusher engines. The maximum efficiency 
in operating pusher engines is obtained when the pusher engine 
is kept constantly at work, and this is facilitated when the pusher 
grade is as long as possible, i.e., when the heavy grades and the 
great bulk of the difference of elevation to be surmounted is 
at one place. For example, a pusher grade of three miles fol- 
lowed by a comparatively level stretch of three miles and then 
by another pusher grade of two miles cannot all be operated as 
cheaply as a continuous pusher grade of five miles. Either 
the two grades must be operated as a continuous grade of eight 
miles (sixteen pusher miles per trip) or else as two short pusher 
grades, in which case there would be a very great loss of time 
and a difficulty in so arranging the schedules that a train need 
not wait for a pusher or the pushers need not waste too much 
time in idleness waiting for trains. If the level stretch were 
imperative, the two grades would probably be operated as one, 
but an effort should be made to bring the grades together. It 
is not necessary to bring the trains to a stop to uncouple the 
pusher engine, but a stop is generally made for coupling on, and 
the actual cost in loss of energy and in wear and tear of stopping 
and starting a heavy train is as great as the cost of running 
an engine light for several miles. 

There are two ways in which it is possible to economize in 
Vhe use of pusher engines, (a) When the traffic of a road is 



538 RAILKOAD CONSTRUCTION. § 448. 

SO very light that a pusher engine will not be kept reasonably 
busy on the pusher grade it may be worth while to place a 
siding long enough for the longest trains both at top and bottom 
of the pusher grade and then take up the train in sections. 
Perhaps the worst objection to this method is the time lost 
while the engine runs the extra mileage, but with such very 
light traffic roads a little time more or less is of small consequence. 
On light traffic roads this method of surmounting a heavy grade 
will be occasionally adopted even if pushers are never used. 
If the traffic is fluctuating, the method has the advantage 
of only requiring such operation when it is needed and avoiding 
the purchase and operation of a pusher engine which has but 
little to do and which might be idle for a considerable proportion 
of the year, {h) The second possible method of economizing 
is only practicable when a pusher grade begins or ends at or 
near a station yard where switching-engines are required. In 
such cases there is a possible economy in utilizing the switching- 
engines as pushers, especially when the work in each class is 
small, and thus obtain a greater useful mileage. But such cases 
are special and generally imply small traffic. 

A telegraph-station at top and bottom of a pusher grade is 
generally indispensable to effective and safe operation. 

449. Length of a pusher grade. The virtual length of the 
pusher grade, as indicated by the mileage of the pusher engine, 
is always somewhat in excess of the true length of the grade 
as shown on the proffie, and sometimes the excess length is 
very great. If a station is located on a lower grade within a 
mile or so of the top or bottom of a pusher grade, it will ordina- 
rily be advisable to couple or uncouple at or near the station, 
since the telegraph-station, switching, and signaling may be 
more economically operated at a regular station. If the extra 
engine is coupled on ahead of the through engine (as is some- 
times required by law for passenger trains) the uncoupling at 
the top of the grade may be accom.plished by running the assist- 
ant engine ahead at greater speed after it is uncoupled, and, 
after running it on a siding, clearing the track for the train. 
But this requires considerable extra track at the top of the grade. 
Therefore, when estimating the length of the pusher grade, 
the most desirable position for the terminal sidings must be 
studied and the length determined accordingly rather than 



§ 451. GRADE. 539 

by measuring the mere length of the grade on the profile. Of 
course these odd distances are always excess; the coupling or 
uncouphng should not be done while on the grade. 

450. The cost of pusher-engine service. When we analyze 
the elements of cost, we will find that many of them are dependent 
only on time, while others are dependent upon mileage. Still 
others are dependent on both. Very much will depend on the 
constancy of the service, and this in turn depends on the train 
schedule and on a variety of local conditions which must be 
considered for each particular case. The effect of a pusher- 
engine on maintenance of way may be considered to be the same 
as the cost of an additional engine to handle a given traffic, as 
developed in § 198. The same total allowance for the expenses 
of maintenance of way (7.45 %) will therefore be made. Although 
the cost of repairs and renewals of engines is evidently a function 
of the mileage, and would therefore be somewhat less for a 
pusher-engine which did Httle work than for an engine which 
was worked to the Hmit of its capacity, yet it is only safe to 
make the same allowance as for other engines. Other items 
of maintenance of equipment are evidently to be ignored. The 
item of w^ages of enginemen will evidently depend upon the 
system employed on the particular road. Whatever the precise 
system the general result is to pay the enginemen as much in 
wages as the average payment for regular service, and therefore 
the full allowance for Item 80 will be made. Similarly we must 
allow the full cost of the items for engine supphes. While the 
engine is doing its heavy work in cHmbing up the grade, the 
consumption of fuel and water is certainly greater than the 
average; but, on the other hand, on the return trip, when the 
engine is running hght, it probably runs for a considerable por- 
tion of the distance actually without steam, and therefore the 
consumption of fuel and water will nearly, if not quite, average 
the consumption for an engine running up and down grade 
along the whole fine. That portion of fuel consumption which 
is due to radiation, blowing-off steam, and the many other 
causes previpusly enumerated, will be the same regardless of 
the work done. We therefore allow 100% for all of these items 
of engine supphes. In general we must add 100% for Items 90, 
91, and 94, the cost of switchmen and telegraphic service. While 
there might be cases where there would be no actual addition 



540 



EAILROAD CONSTRUCTION. 



§451. 



TABLE XXIX. — COST FOR EACH MILE OF PUSHER-ENGINE 

SERVICE. 



Item 
number. 


Item (abbreviated). 


Normal 
average. 


Per cent 
affected. 


Cost per 
engine 
mile, 

per cent. 


2-6,9 
25-27 


Track material, labor, bridges. . 
Steam locomotives 


14.89% 
8.61 

7.80 
11.41 

1.17 


50 
100 

100 
100 

100 


7.45 
8.61 


80.81 


Road enginemen and engine- 
house expenses 


7.80 


82-85 

90, 91, 

^94 


Fuel and other engine supplies . . 
/ Signaling, flagmen, and tele- 
\ graph 


11.41 
1.17 










45.05 . 











to the pay-rolls or the operating expenses on account of these 
items, we are not justified in general in neglecting to add the 
full quota for such service. Collecting these items we will have 
45.05% of the average cost of a train-mile for the cost of each 
mile run by the pusher engine. Using the same figure as before, 
$1.50, for the cost of one train mile, we have 67.57 c. for each 
mile run. 

451. Numerical comparison of pusher and through grades. 
In § 445 the computation was made of the desirabihty of re- 
ducing a 1.6% ruhng grade to a 1.2% grade. Suppose it is 
found that by keeping the 1.6% grades as pusher grades having 
a total length of 20 miles on a 100-mile division, the other grades 
may be reduced to a grade not exceeding 0.713% (the correspond- 
ing through grade) for an expenditure of $200,000. Will it 
pay? The saving by cutting down trains from 8 to 4, computed 
as before, would be (see § 445), 4X10.6245X100X365 =$91,177. 
But this saving is only accomplished by the employment of 
pushers making four round trips over 20 miles of pusher grades 
at a cost of 4 X20X2X365X$0.6757 = $39,461. 

The net annual saving is therefore $51,716, which when 
capitalized at 5% =$1,034,320. 

The above estimate probably has this defect. The total 
daily pusher-engine mileage is but 2X4X20 = 160, scarcely 
work enough for two pushers. Unless the pusher grades were 
bunched into two groups of about 10 miles each, two pusher 
engines could not do the work. If the number of trains was 



§ 452. GKADE. 541 

much larger, then the above method of calculation would be 
more exact even though the 20 miles of pusher grade was divided 
among four or five different grades. Therefore with the above 
data the annual cost of the pusher service would probably be 
much more, say, $60,000, and the annual saving about $30,000, 
which would justify an expenditure of $600,000. But even this 
would very amply justify the assumed expenditure of $200, 000 
which would accomphsh this result. 

The above computation is but an illustration of the general 
truth which has been previously stated. In spite of the un- 
certainties and the variations of many items in the above esti- 
mates it will generally be possible to make a computation which 
will show unquestionably, as in the above instance, what is 
the best and the most economical method of procedure. ^\lien 
the capitalized valuations of both methods are so nearly equal 
that a proper choice is more difficult, the question will frequently 
be determined by the relative ease of raising additional capital. 

BALANCE OP GRADES FOR UNEQUAL TRAFFIC. 

452. Nature of the subject. It sometimes happens, as when 
a road runs into a moimtainous country for the purpose of 
hauling therefrom the natural products of lumber or minerals, 
that the heavy grades are all in one direction — that the whole 
line consists of a more or less unbroken climb having perhaps 
a few comparatively level stretches, but no down grade (except 
possibly a slight sag) in the direction of the general up grade. 
With such lines this present topic has no concern. But the 
majority of railroads have termini at nearly the same level 
(500 feet in 500 miles has no practical effect on grade) and 
consist of up and down grades in nearly equal amounts and 
rates. The general rate of ruling grade is determined by the 
character of the country and the character and financial backing 
of the road to be built. It is always possible to reduce the grade 
at some point by "development" or in general by the expen- 
diture of more money. It has been tacitly assumed in the 
previous discussions that w^hen the ruling grade has been de- 
termined all grades in either direction are cut down to that 
limit. If the traffic in both directions were the same this would 
be the proper policy and sometimes is so. But it has developed, 



542 EAILROAD CONSTRUCTlOHo § 452. 

especially on the great east and west trunk lines, that the weight 
of the eastbound freight traffic is enormously greater than that 
of the westbound — that westbound trains consist very largely of 
'^ empties'' and that an engine which could haul twenty loaded 
cars up a given grade in eastbound traffic could haul the same 
cars empty up a much higher grade when running west. As 
an illustration of the large disproportion which may exist, the 
eastbound ton-mileage on the P„ R. R. between the years 1851 
and 1885 was 3.7 times the westbound ton-mileage. Between 
the years 1876 and 1880 the ratio rose to more than 4.5 to 1. 
On such a basis it is as important and necessary to obtain, say, 
a 0.6% ruling grade against the eastbound traffic as to have, 
say, a 1.0% grade against the westbound traffic. This is the 
basis of the following discussion. It now remains to estimate 
the probable ratio of the traffic in the two directions and from 
that to determine the proper "balance" of the opposite ruling 
grades. 

453. Computation of the theoretical balance. Assume first, 
for simplicity, that the exact business in either direction is 
accurately known. A little thought will show the truth of the 
following statements. 

1. The locomotive and passenger-car traffic in both directions 
is equal. 

2. Except as a road may carry emigrants, the passenger 
traffic in both directions is equal. Of course there are innumer- 
able individual instances in which the return trip is made by 
another route, but it is seldom if ever that there is any marked 
tendency to uniformity in this. Considering that a car load 
of, say, 50 passengers at 150 pounds apiece weigh but 7500 
pounds, which is \ of the 45000 pounds which the car may 
weigh, even a considerable variation in the number of passengers 
will not appreciably affect the hauling of cars on grades. On 
parlor-cars and sleepers the ratio of live load to dead load (say 
20 passengers, 3000 pounds, and the car, 75000 pounds) is 
even more insignificant. The effect of passenger traffic on 
balance of grades may therefore be disregarded. 

3. Empty cars have a greater resistance per ton than loaded 
cars. Therefore in computing the hauling capacity of a loco- 
motive hauling so many tons of "empties," a larger figure must 
be used for the ordinary tractive resistances — say four pounds 
per ton greater. 



§ 453. GRADE. 543 

4, Owing to greater or less imperfections of management a 
small percentage of cars will run empty or but partly full in 
the direction of greatest traffic. 

5, Freight having great bulk and weight (such as grain, 
lumber, coal, etc.) is run from the rural districts toward the 
cities and manufacturing districts. 

6, The return traffic — manufactured products — although worth 
as much or more, do not weigh as much. 

As a simple numerical illustration assume that the weight 
of the cars is ^ and the live load f of the total load when 
the cars are "full" — although not loaded to their absolute 
limit of capacity. Assume that the relative weight of live load 
to be hauled in the other direction is but J; assume that the 
grade against the heaviest traffic is 0.9%. Since the tractive 
resistance per ton is considerably greater in the case of imloaded 
cars than it is in the case of loaded cars, allowance must be 
made for this in calculating the train resistance. Mr. A. C. 
Dennis, of the Canadian Pacific Railway Company, has made 
some elaborate tests of train resistance for trains which were 
alternately loaded and empty, and found that the tractive resist- 
ance of loaded cars was very uniform at 4.7 pounds per ton- 
when the weight of the empty cars was J of the total woight. 
He also found that the tractive resistance of empty cars was 
very imiform at 8.9 pounds per ton. Although the live load 
capacity of a box-car is usually considerably more than twice 
the weight of the empty car, it will probably coincide more 
nearly with actual running conditions to consider that the live 
load is just twice the dead load. Assume that these loads are 
being hauled by a consolidation engine with a total weight, 
including engine and tender, of 107 tons, of which 106000 pounds 
is on the drivers. We will assume that the tractive resistance 
of the locomotive is likewise 4.7 pounds per ton. On the 0.9% 
grade, the grade resistance will be 18 pounds per ton, and there- 
fore the total resistance is 22.7 pounds per ton. Assume that 
this engine is working with a tractive adhesion of J; the trac- 
tive power at the circumference of the drivers will be J of 
106000 pounds, or 26500 pounds. Dividing this by 22.7, we 
obtain 1167 as the gross load of the train in tons. Subtracting 
the weight of the locomotive, 107 tons, we have 1060 tons as 
the weight of the loaded cars which could be hauled by this 
locomotive up a 0.9% grade, assuming an adhesion of J. Since 



544 EAILROAD CONSTRUCTION. ^ § 453. 

the traffic in the other direction is but J, we will assume that 
f of the return cars are empty. We then have 353 tons of 
loaded cars with a locomotive weighing 107 tons, and 236 tons 
of empty cars in the return train. The loaded cars with the 
locomotive will weigh 460 tons, and their tractive resistance 
will be 4.7 pounds per ton, or 2162 pounds. The 236 tons of 
empty cars will have a resistance of 8.9 pounds per ton, or a 
total tractive resistance of 2100 pounds. This makes a total 
of 4262 pounds of tractive resistance. Subtracting this from 
the 26500 of total adhesion of the drivers, we have left 22238 
as the amount of pull available for grade. But the return train 
weighs 696 tons. Dividing this into 22238, we find that 32 
pounds per ton is available for grade, which is the resistance on 
a 1.60% grade. Therefore, under the above conditions^ a 0.9% 
grade against the heaviest traffic will correspond with a 1.60% 
grade against the lighter traffic. 

Of course these figures will be slightly modified by variations 
in the assumptions as to the tractive resistance of loaded and 
unloaded cars, and more especially by variations in the ratio 
of live load to dead load in the two directions. Therefore no 
great accuracy can be claimed for the ratio of these two grades 
in opposite directions, nevertheless the above calculation shows 
unmistakably that under the given conditions, a very consider- 
able variation in the rate of grade in opposite directions is not 
on!y justifiable, but a neglect to allow for it would be a great 
economic error. 

454. Computation of relative traffic. Some of the principal 
elements have already been referred to, but in addition the 
following facts should be considered. 

(a) The greatest disparity in traffic occurs through the hand- 
ling of large amounts of coal, lumber, iron ore, grain, etc. On 
roads which handle but little of these articles or on which for 
local reasons coal is hauled one way and large shipments of 
grain the other way the disparity will be less and will perhaps 
be insignificant. 

(6) A marked change in the development of the country iray, 
and often does, cause a marked difference in the disparity of 
traffic. The heaviest traffic (in mere weight) is always toward 
manufacturing regions and away from agricultural regions. But 
when a region, from being purely agricultural or mineral, be- 
comes largely manufacturing, or when a manufacturing region 



§ 454, GRADE. 545 

develops an industry which will cause a growth of heavy freight 
traffic from it, a marked change in the relative freight movement 
will be the result. 

(c) Very great fluctuations in the relative traffic may be 
expected for prolonged intervals. 

{d) An estimate of the relative traffic may be formed by 
the same general method used in computing the total traffic 
of the road (see § 373, Chap. XIX) or by noting the relative 
traffic on existing roads which may be assumed to have practically 
the same traffic as the proposed road will obtain. 



CHAPTER XXIV. 

THE IMPROVEMENT OF OLD LINES. 

455. Classification of improvements. The improvements here 
considered are only those of ahgnment — horizontal and vertical. 
Strictly there is no definite limit, either in kind or magnitude, 
to the improvements which may be made. But since a railroad 
cannot ordinarily obtain money, even for improvements^ to 
an amount greater than some small proportion of the pre- 
viously invested capital, it becomes doubly necessary to expend 
such money to the greatest possible advantage. It has been 
previously shown that securing additional business and increas- 
ing the train load are the two most important factors in increas- 
ing dividends. After these, and of far less importance, come 
reductions of curvature, reductions of distance (frequently of 
doubtful policy, see Chap. XXI, §414), and elimination of sags 
and humps. These various improvements will be briefly dis- 
cussed. 

(a) Securing additional business. It is not often possible 
by any small modification of alignment to materially increase 
the business of a road. The cases which do occur are usually 
those in which a gross error of judgment was committed during 
the original construction. For instance, in the early history 
of railroad construction many roads were largely aided by the 
towns through which the road passed, part of the money neces- 
sary for construction being raised by the sale of bonds, wliich 
were assumed or guaranteed and subsequently paid by the 
toVns. Such aid was often demanded and exacted by the 
promoters. Instances are not unknown where a failure to 
come to an agreement has caused the promoters to deliberately 
pass by the town at a distance of some miles, to the mutual 
disadvantage of the road and the town. If the town subsequent- 
ly grew in spite of this disadvantage, the annual loss of business 
might readily amount to more than the original sum in dispute. 

546 



§ 457. IMPRO'/EMENT OF OLD LINES. 547 

Such an instance would be a legitimate opportunity for study 
of the advisability of a re-location. 

As another instance (the original location being justifiable) 
a railroad might have been located along the bank of a consider- 
able river too wdde to be crossed except at considerable expense. 
When originally constructed the enterprise would not justify 
the two extra bridges needed to reach the town. A growth in 
prosperity and in the business obtainable might subsequently 
make such extra expense a profitable investment. 

(b) Increasing the train load. On account of its importance 
this mil be separately considered in § 458 et seq. 

(c) Reductions in curvature and distance and the elimination 
of sags and humps. The financial value of these improvements 
has already been discussed in Chapters XXI, XXII, and XXIII. 
Such improvements are constantly being made by all progressive 
roads. The need for such changes occurs in some cases because 
the original location was very faulty, the revised location being 
no more expensive than the original, and in other cases because 
the original location was the best that was then financially 
possible and because the present expanded business wiU justify 
a change. 

(d) Changing the location of stations or of passing sidings. 
The station may sometimes be re-located so as to bring it nearer 
to the business center and thus increase the business done. 
But the principal reasons for re-locating stations or passing 
sidings is that starting trains may have an easier grade on which 
to overcome the additional resistances of starting. Such changes 
will be discussed in detail in § 460. 

456. Advantages of re-locations. There are certain undoubted 
advantages possessed by the engineer who is endeavoring to 
improve an old line. 

(a) The gross traffic to be handled is definitely known. 

(h) The actual cost per train-mile for that road (which may 
differ very greatly from the average) is also known, and therefore 
the value of the proposed improvement can be more accurately 
determined. 

. (c) The actual performance of such locomotives as are used 
on the road may be studied at leisure and more reliable data 
may be obtained for the computations. 

457. Disadvantages of re-locations. The disadvantages are. 
generally more apparent and frequently appear practically 



548 RAIIiPtOAD CONSTRUCTION. § 457. 

insuperable — more so than they prove to be on closer inspection, 
(a) It frequently means the abandonment of a greater or less 
length of old line and the construction of new line. At first 
thought it might seem as if a change of line such as would permit 
an increase of train-load of 50 or perhaps 100% could never 
be obtained, or at least that it could not be done except at an 
impracticable expense. On the contrary a change of 10% 
of the old line is frequently all that is necessary to reduce the 
grades so that the train-loads hauled by one engine may be 
nearly if not quite doubled. And when it is considered that 
the cost of a road to sub-grade is generally not more than one- 
third of the total cost of construction and equipment per mile, 
it becomes plain that an expenditure of but a small percentage 
of the original outlay, expended where it will do the most good, 
will often suffice to increase enormously the earning capacity. 

(b) One of the most difficult matters is to convince the finan- 
cial backers of the road that the proposed improvement will 
be justifiable. The cause is simple. The disadvantages of the 
original construction lie in the large increase of certain items 
of expense which are necessary to handle a given traffic. And 
yet the fact that the expenditures are larger than they need 
be are only apparent to the expert, and the fact that a saving 
may be made is considered to be largely a matter of opinion 
until it is demonstrated by actual trial. On the other hand 
the cost of the proposed changes is definite, and the very fact 
that the road has been uneconomically worked and is in a poor 
financial condition makes it difficult to obtain money for im- 
provements. 

(c) The legal right to abandon a section of operated line 
and thus reduce the value of some adjoining property has 
sometimes been successfully attacked. A common instance 
would be that of a factory which was located adjoining the right 
of way for convenience of transportation facilities. The abandon- 
ment of that section of the right of way would probably be fatal 
to the successful operation of the factory. The objection may 
be largely eliminated by the maintenance of the old right of 
way as a long siding (although the business of the factory might 
not be worth it), but it is not always so easy of solution, and 
this phase of the question must always be considered. 



§ 458, IMPROVEMENT OF OLD LINES. 549 



REDUCTION OF VIRTUAL GRADE. 

458. Obtaining data for computations. As developed in the 
last chapter (§§ 432-434) the real object to be attained is the 
reduction of the virtual grade. The method of comparing grades 
imder various assumed conditions was there discussed. When 
the road is still ''on paper'' some such method is all that is 
possible; but when the road is in actual operation the virtual 
grade of the road at various critical points, Tvdth the rolling 
stock actually in use, may be determined by a simple test and 
the effect of a proposed change may be reliably computed. 
Bearing in mind the general principle that the virtual grade 
line is the locus of points determined by adding to the actual 
grade profile ordinates equal to the velocity head of the train, 
it only becomes necessary to measure the velocity at various 
points. Since the velocity is not usually uniform, its precise 
determination at any instant is almost impossible, but it will 
generally be found to be sufficiently precise to assume the velocity 
to be imiform for a short distance, and then observe the time 
required to pass that short space. Suppose that an ordinary 
watch is used and the time taken to the nearest second. At 
30 miles per hour, the velocity is 44 feet per Second. To obtain 
the time to wdthin 1%, the time would need to be 100 seconds 
and the space 4400 feet. But with variable velocity there 
would be too great error in assuming the velocity as uniform 
for 4400 feet or for the time of 100 seconds. Using a stop- 
watch registering fifths of a second, a 1% accuracy would 
require but 20 seconds and a space of 880 feet, at 30 miles per 
hour. Wellington suggests that the space be made 293 feet 
4 inches, or -^^ of a mile; then the speed in miles per hour 
equals 200 -r- s, in which s is the time in seconds required to 
traverse the 293' 4". For instance, suppose the time required 
to pass the interval is 12.5 seconds. -^-^ mile in 12.5 seconds = 
one mile in 225 seconds, or 16 miles per hour. But likemse 
200-^12.5=16, the required velocity. The following features 
should be noted when obtaining data for the computations: 

(a) All critical grades on the road should be located and 
their profiles obtained — by a survey if necessary. 

(h) At the bottom and top of all long grades (and perhaps at 
intermediate points if the grades are very long) spaces of known 



550 RAILROAD CONSTRUCTION, § 458. 

length (preferably 293 J feet) should be measured off and marked 
by flags, painted boards, or any other serviceable targets, 

(c) Provided with a stop-watch marking fifths of seconds 
the observer should ride on the trains affected by these grades 
and note the exact interval of time required to pass these spaces. 
If the space is 293 J feet, the velocity in miles per hour =200 -r- 
interval in seconds. In general, 

^ _ distance in feet X 3600 
time in seconds X 5280* 

(d) Since these critical grades are those which require the 
greatest tax on the power of the locomotive, the conditions 
under which the locomotive is working must be known — ^i.e., 
the steam pressure, point of cut-off, and position of the throttle. 
Economy of coal consumption as well as efficient w^orking at 
high speeds requires that steam be used expansively (using an 
early cut-off), and even that the throttle be partly closed; but 
when an engine is slowly climbing up a maximum grade with a 
full load it is not exerting its maximum tractive power unless 
it has its maximum steam pressure, wide-open throttle, and is 
cutting off nearly at full stroke. These data must therefore 
be obtained so as to know whether the engine is developing 
at a critical place all the tractive force of which it is capable. 
The condition of the track (wet and slippery or dry) and the 
approximate direction and force of the wind should be noted 
with sufficient accuracy to judge whether the test has been made 
under ordinary conditions rather than under conditions which 
are exceptionally favorable or unfavorable. 

(e) The train-loading should be obtained as closely as possible. 
Of course the dead weight of the cars is easily found, and the 
records of the freight department will usually give the live 
load with all sufficient accuracy. 

459. Use of the data obtained. A very brief inspection 
of the results, freed from refined calculations or uncertainties, 
will demonstrate the following truths: 

(a) If, on a uniform grade, the velocity increases, it shows 
that, under those conditions of engine working, the load is less 
than the engine can handle on that grade 

(b) If the velocity decreases, it shows that the load is greater 
than the engine can handle on an indefinite length of such 



§ 459. . IMPROVEMENT OF OLD LINES. 551 

grade. It shows that such a grade is being operated by momen- 
tum. Frciii the rate of decrease of velocity the maximum 
practicable length of such a grade (starting mth a given velocity) 
may be easily computed. 

(c) By combining results under different conditions of grade 
but with practically the same engine working, the tractive 
power of the engine may be determined (according to the prin- 
ciples pre^dously demonstrated) for any grade and velocity. 
For example: On an examination of the profile of a division 
of a road the maximum grade was found to be 1.62% (85.54 
feet per mile). At the bottom and near the top of this grade 
two lengths of 293' 4'' are laid off. The distance between the 
centers of these lengths is 6000 feet. A freight train moving 
up the grade is timed at 9f seconds on the lower stretch and 7f 

seconds on the upper. These times correspond to ^— and ;=— ^ 

or 21.3 and 26.3 miles per hour respectively. It is at once 

observed that the velocity has increased and that the engine 

could draw even a hea^der load up such a grade for an indefinite 

distance. How much heavier might the load be? 

For simplicity we will assume that the conditions were 

normal, neither exceptionally favorable nor unfavorable, and 

that the engine was worked to its maximum capacitj^. The 

engine is a ^' consolidation" w^eighing 128700 pounds, ^ith 

112600 pounds on the drivers. The train-load behind the 

engine consists of ten loaded cars weighing 465 tons and eleven 

empties weighing 183 tons, thus making a total train- weight of 

712 tons. Applying Eq. 140, we find that the additional force 

which the engine has actually exerted per ton in increasing the 

velocity from 21.3 to 26.3 miles per hour in a distance of 6000 

feet is 

70 224 
P- -g^^(26. 32-21. 32) =2.78 pounds per ton 

The grade resistance on a 1.62% grade is 32.4 pounds per 
ton. The average train resistance may be computed similarly 
to the method adopted in § 439: 

465 1 

tons at 4 .7 pounds per ton = 2486 pounds 

183 '' '' 8.9 '' '' '' =1629 '' 
tt ^ 

712 4115 '' 



552 RAILROAD CONSTRUCTION. 

The average tractive resistance is therefore 4115-5-712 = 5.78 
pounds per ton. Adding the grade resistance (32.4) we have 
a total train resistance of 38.18 pounds per ton. But, com- 
puting from the increase in velocity, the locomotive is evidently- 
exerting a pull of 2.78 pounds per ton in excess of the computed 
required pull on that grade, or a total pull of 40.96 pounds 
per ton. Therefore the train load might have been increased 
proportionately and might have been made 



^,^^2.78 + 38.18 ^- . . 
712 X — 28"jg — = 764 tons. 



This shows that 52 tons additional might have been loaded 
on to the train, or say, three more empties or one additional 
loaded car. 

A pull of 40.96 pounds per ton means a total adhesion at the 
drivers of 29164 pounds, which is about 26% of the weight on 
the drivers — 1126C0 pounds. This indicates average condi- 
tions as to traction, although better conditions than can be 
depended on for regular service. 

The above calculation should of course be considered simply 
as a '^single observation. '' The performance of the same engine 
on the same grade (as well as on many other grades) on succeed- 
ing days should also be noted. It may readily happen that 
variations in the condition of the track or of the handling of the 
engine may make consideralDle variation in the results of the 
several calculations, but when the work is properly done it is 
always possible to draw definite and very positive deductions. 

460. Reducing the starting grade at stations. The resistance 
to starting a train is augmented from two causes : (a) the trac- 
tive resistances are usually about 20 pounds per ton instead 
of, say, 6 pounds, and (b) the inertia resistance must be overcome. 
The inertia resistance of a freight train (see § 347) which is 
expected to attain a velocity of 15 miles per hour in a distance 
of 1000 feet is (see Eq. 140) 

70 224 
p = il!_^(i52 _ 0) = 15 . 8 pounds per ton, which is the equiva- 
lent of a 0.79% grade. Adding this to a grade which nearly or 
quite equals the ruling grade, it virtually creates a new and 
higher ruling grade. Of course that additional force can be 
greatly reduced at the expense of slower acceleration, but even 



§460. 



IMPROVEMENT OF OLD LINES. 



553 



this cannot be done indefinitely, and an acceleration to only 
15 miles per hour in 1000 feet is as slow as should be allowed 
for. With perhaps 14 pounds per ton additional tractive 
resistance, we have about 30 pounds per ton additional — equiva- 




Fig. 217. 



lent to a 1.5% grade. Instances are known where it has proven 
wise to create a hump (in what was otherwise a uniform grade) 
at a station. The effect of this on high-speed passenger trains 
moving wp the grade would be merely to reduce their speed 
Yery slightly. No harm is done to trains moving down the 
grade. Freight trains moving up the grade and intending to 
stop at the station will merely have their velocity reduced as 
they approach the station and will actually save part of the 
wear and tear otherwise resulting from applying brakes. When 
the trains start they are assisted by the short down grade, 
just where they need assistance most. Even if the grade CD 
is still an up grade, the pull required at starting is less than that 
required on the uniform grade by an amount equal to 20 times 
the difference of the grade in per cent. 



APPENDIX. 

THE ADJUSTMENTS OF mSTEUMEKTS. ' 

The accuracy of instrumental work may be vitiated by any 
one of a large number of inaccuracies in the geometrical relations 
of the parts of the instruments. Some of these relations are so 
apt to bf^ pltered by ordinary usage of the instrument that the 
makers have provided adjusting-screws so that the inaccuracies 
may be readily corrected. There are other possible defects, 
which, however, will seldom be found to exist, provided the 
instrument was properly made and has never been subjected tc 
treatment sufficiently rough to distort it. Such defects, when 
found, can only be corrected by a competent instrument-maker 
or repairer. 

A WARNING is necessary to those who would test the accuracy 
of instruments, and especially to those whose experience in such 
work is small. Lack of skill in handling an instrument will 
often indicate an apparent error of adjustment when the real 
error is very different or perhaps non-existent. It is always a 
safe plan when testing an adjustment to note the amount of the 
apparent error; then^ beginning anew, make another independent 
determination of the amount of the error. When two or more 
"perfectly independent determinations of such an error are made 
it will generally be found that they differ by an appreciable 
amount. The differences may be due in variable measure to 
careless inaccurate manipulation and to instrumental defects < 
which are wholly independent of the particular test being made. 
Such careful determinations of the amounts of the errors are 
generally advisable in view of the next paragraph. 

Do NOT DISTURB THE ADJUSTING-SCREWS ANY MORE THAN 

NECESSARY. Although metals are apparently rigid, they are 
really elastic and yielding. If some parts of a complicated 
mechanism, which is held together largely by friction, are sub- 
jected to greater internal stresses than other parts of the mech- 

554 



APPENDIX. 555 

anism, the jarring resulting from handling will frequently cause 
a slight readjustment in the parts which will tend to more nearly 
equalize the internal stresses. Such action frequently occurs 
with the adjusting mechanism of instruments. One screw may 
be strained more than others. The friction of parts may pre- 
vent the opposing screw from immediately taking up an equal 
stress. Perhaps the adjustment appears perfect under these 
conditions Jarring diminishes the friction between the parts, 
and the unequal stresses tend to equalize. A motion takes place 
which, although microscopically minute, is sufficient to indicate 
an error of adjustment. A readjustment made by unskillful 
hands may not make the final adjustment any more perfect. 
The frequent shifting of adjusting-screws wears them badly, 
and when the screws are worn it is still more difficult to keep 
them from mo\dng enough to vitiate the adjustments. It is 
therefore preferable in many cases to refrain from disturbing the 
adjusting-screws, especially as the accuracy of the work done is 
not necessarily affected by errors of adjustment, as may be 
illustrated : 

(a) Certain operations are absolutely unaffected by certain 
errors of adjustment. 

(&) Certain operations are so slightly affected by certain small 
errors of adjustment that their effect may properly be neglected. 

(c) Certain errors of adjustment may be readily allowed for 
and neutralized so that no error results from the use of the 
imadjusted" instrument. Illustrations of all these cases will be 
given under their proper heads. 

ADJUSTMENTS OF THE TRANSIT. 

1. To have the plate-huhhles in the center of the tubes when the 
axis is vertical. Clamp the upper plate and, with the lower 
clamp loose, swing the instrument so that the plate-bubbles are 
parallel to the lines of opposite leveling-screws. Level up until 
both bubbles are central. Swing the instrument 180°. If the 
bubbles again settle at the center, the adjustment is perfect. If 
either bubble does not settle in the center, move the leveling- 
screws until the bubble is half-way back to the center. Then, 
before touching the adjusting-screws, note carefully the position 
of the bubbles and observe whether the bubbles always settle at 
the same place in the tube, no matter to what position the in- 



556 RAILROAD CONSTRUCTION". 

stniment may be rotated. When the instrument is so leveled, 
the axis is truly vertical and the discrepancies between this 
constant position of the bubbles and the centers of the tubes 
measure the errors of adjustment. By means of the adjusting- 
screws bring each bubble to the center of the tube. If this is 
done so skillfully that the true level of the instrument is not 
disturbed, the bubbles should settle in the center for all positions 
of the instrument. Under unskillful hands, two or more such 
trials may be necessary. 

When the plates are not horizontal, the measured angle is greater than 
the true horizontal angle by the difference between the measured angle 
and its projection on a horizontal plane. When this angle of inclination 
is small, the difference is insignificant. Therefore when the plate-bubbles 
are very nearly in adjustment, the error of measurement of horizontal 
angles may be far within the lowest unit of measurement used. A small 
error of adjustment of the plate-bubble 'perpendicular to the telescope will 
affect the horizontal angles by only a small proportion of the error, which 
will be perhaps imperceptible. Vertical angles will be affected by the' 
same insignificant amount. A small error of adjustment of the plate- j 
bubble parallel to the telescope will affect horizontal angles very slightly, ^ 
but will affect vertical angles by the full amount of the error. 

All error due to unadjusted plate-bubbles may be avoided by noting in 
what positions in the tubes the bubbles will remain fixed for all positions 
of azimuth and then keeping the bubbles adjusted to these positions, for 
the axis is then truly vertical. It will often save time to work in this way 
temporarily rather than to stop to make the adjustments. This should 
especially be done when accurate vertical angles are required. 

When the bubbles are truly adjusted, they should remain stationary 
regardless of whether the telescope is revolved with the upper plate loose 
and the lower plate clamped or whether the whole instrument is revolved, 
the plates being clamped together. If there is any appreciable difference, 
it shows that the two vertical axes or ** centers" of the plates are not con- 
centric. This may be due to cheap and faulty construction or to the exces- 
sive wear that may be sometimes observed in an old instrument originally 
well made. In either case it can only be corrected by a maker. 

2. To make the revolving axis of the telescope perpendicular to 
the vertical axis of the instrument. This is best tested by using 
a long plumb-line, so placed that the telescope must be pointed 
upward at an angle of about 45° to sight at the top of the plumb- 
line and downward about the same amount, if possible, to 
sight at the lower end. The vertical axis of the transit must 
be made truly vertical. Sight at the upper part of the line; 
clamping the horizontal plates. Swing the telescope down 
and see if the cross-wire again bisects the cord. If so, the 
adjustment is probably perfect (a conceivable exception will be 



APPENDIX. 557 

noted later); if not, raise or lower one end of the axis by mtang 
of the adjusting-screws, placed at the top of one of the standards, 
until the cross-wire will bisect the cord both at top and bottom. 
The plumb-bob may be steadied, if necessary, by hanging it 
in a pail of water. As many telescopes cannot be focused 
on an object nearer than G or 8 feet from the telescope, this 
method requires a long plumb-line swung from a high point, 
which may be inconvenient. 

Another method is to set up the instrument about 10 feet 
from a high wall. After leveling, sight at some convenient 
mark high up on the wall. Swing the telescope down and make 
a mark (when working alone some convenient natural mark may 
generally be found) low down on the wall. Plunge the telescope 
and revolve the instrument about its vertical axis and again sight 
at the upper inark. Swing down to the lower mark. If the 
wire again bisects it, the adjustment is perfect. If not, fix a 
point half-way between the two positions of the lower mark. 
The plane of this point, the upper point, and the center of the 
instrument is truly vertical. Adjust the axis to these upper and 
lower points as when using the plumb-line. 

3. To make the line of collimation perpendicular to the revolving 
axis of the telescope. With the instrument level and the telescope 
nearly horizontal point at some well-defined point at a distance 
of 200 feet or more. Plunge the telescope and estabhsh a point 
in the opposite direction. Turn the whole instniment about the 
vertical axis until it again points at the first mark. Again 
plunge to '' direct position" {i.e., with the level-tube under 
the telescope). If the vertical cross- wire again points at the 
second mark, the adjustment is perfect. If not, the error is 
one-fourth of the distance between the two positions of the 
second mark. Loosen the capstan screw on one side of the 
telescope and tighten it on the other side until the vertical 
wire is set at the one-fourth mark. Turn the whole instrument 
by means of the tangent screw until the vertical wire is midway 
between the two positions of the second nmrk. Plunge the 
telescope. If the adjusting has been skillfully done, the cross- 
wire should come exactly to the first mark. As an ''erecting 
eyepiece" reinverts an image already inverted, the ring carr^dng 
the cross-wdres must be moved in the same direction as the 
apparent error in order to correct that error. 



558 RAILROAD CONSTRUCTION. 

The necessity for the third adjustment lies principally in the practice 
of producing a line by plunging the telescope, but when this is required to 
be done with great accuracy it is always better to obtain the forward point 
by reversion (as described above for making the test) and take the mean 
of the two forward points. Horizontal and vertical angles are practically 
unaffected by small errors of this adjustment, unless, in the case of hori- 
zontal angles, the vertical angles to the points observed are very different. 

Unnecessary motion of the adjusting-screws may sometimes be avoided 
by carefully establishing the forward point on line by repeated reversions 
of the instrument, and thus determining by repeated trials the exact amount 
of the error. Differences in the amount of error determined would be 
evidence of inaccuracy in manipulating the instrument, and would show 
that an adjustment based on the first trial would probably prove unsatis- 
factory. 

The 2d and 3d adjustments are mutually dependent. If either adjust- 
ment is badly out, the other adjustment cannot be made except as follows^ 

(a) The second adjustment can be made regardless of the third when 
the lines to the high point and the low point make equal angles with the 
horizontal. 

(6) The third adjustment can be made regardless of the second when- 
the front and rear points are on a level with the instrument. 

When both of these requirements are nearly fulfilled, and especially 
when the error of either adjustment is small, no trouble will be found in 
perfecting either adjustment on account of a small error in the other ad- 
justment. 

If the test for the second adjustment is made by means of the plumb- 
line and the vertical cross-wire intersects the line at all points as the tele- 
scope is raised or lowered, it not only demonstrates at once the accuracy 
of that adjustment, but also shows that the third adjustment is either 
perfect or has so small an error that it does not affect the second. | 

4. To have the bubble of the telescope-level in the center of the 
tube when the line of collimation is horizontal. The line of colli- 
mation should coincide with the optical axis of the telescope. 
If the object-glass and eyepiece have been properly centered, 
the previous adjustment will have brought the vertical cross- 
wire to the center of the field of view. The horizontal cross- 
\nre should also be brought to the center of the field of view, 
and the bubble should be adjusted to it. 

a. Peg method. Set up the transit at one end of a nearly 
level stretch of about 300 feet. Clamp the telescope with its 
bubble in the center. Drive a stake vertically under the eye- 
piece of the transit, a,nd another about 300 feet away. Observe 
the height of the center of the ej^epiece (the telescope being 
level) above the stake (calling it a) ; observe the reading of the 
rod when held on the other stake (calling it b) ; take the instru- 
ment to the other stake and set it up so that the eyepiece is 



fi 



APPENDIX. 559 

vertically over the stake, observing the height, c ; take a reading 
on the first stake, calling it d. If this adjustment is perfect, 
then 

a—d — h — Cf 
or {a-d)-{h-c)=0, 
CaU (a-c^)-(6-c)=2m. 
When m is positive, the line points downward; 
" m '^ negative, '' " " upward. 

To adjust: if the line points ujpj sight the horizontal cross- 
wire (by moving the vertical tangent screw) at a point which is 
m lower, then adjust the bubble so that it is in the center. 

By taking several independent values for a, b, c, and d, a mean value 
for m is obtained, which is more reliable a,nd which may save much un- 
necessary working of the adjusting-screws. 

h. Using an auxiliary level. When a carefully adjusted level 
is at hand, this adjustment may sometimes be more easily 
made by setting up the transit and level, so that their lines of 
coUimation are as nearly as possible at the same height. If a 
point may be found which is half a mile or more away and 
which is on the horizontal cross-wire of the level, the horizontal 
cross-wire of the transit may be pointed directly at it, and the 
bubble adjusted accordingly. Any slight difference in the 
heights of the lines of coUimation of the transit and level (say 
y) may almost be disregarded at a distance of J mile or more, 
or, if the difference of level would have an appreciable effect, 
even this may be practically eliminated by making an estimated 
allowance when sighting at the distant point. Or, if a distant 
point is not available, a level-rod with target may be used at a 
distance of (say) 300 feet, making allowance for the carefully 
determined difference of elevation of the two lines of coUimation. 

5. Zero of vertical circle. When the line of coUimation is truly 
horizontal and the vertical axis is truly vertical, the reading 
of the vertical circle should be 0°. If the arc is adjustable, 
it should be brought to 0°. If it is not adjustable, the index 
error should be observed, so that it may be applied to aU readings 
of vertical angles. 

ADJUSTMENTS OF THE WYE LEVEL. 

1. To make the line of coUimation coincide tvith the center of 
the rings. Point the intersection of the cross-wires at some 



560 EAILROAD CONSTRUCTION. 

I^rell-defined point which is at a considerable distance. The in- 
strument need not be level, which allows much greater liberty 
in choosing a convenient point. The vertical axis should be 
clamped, and the clips over the wyes should be loosened and 
raised. Rotate the telescope in the wyes. The intersection of 
the cross- wires should be continually on the point. If it is not^ 
it requires adjustment. Rotate the telescope 180"^ and adjust 
one-half of the error by means of the capstan-headed screws that 
move the cross-wire ring. It should be remembered that, with 
an erecting telescope, on account of the inversion of the image, 
the ring should be moved in the direction of the apparent error. 
Adjust the other half of the error with the leveling-screws. 
Then rotate the telescope 90° from its usual position, sight 
accurately at the point, and then rotate 180° from that position 
and adjust any error as before. It may require several trials, 
but it is necessary to adjust the ring until the intersection of 
the cross-wires will remain on the point for any position of 
rotation. 

If such a test is made on a very distant point and again on a point only 
10 or 15 feet from the instrument, the adjustment may be found correct 
for one point and incorrect for the other. This indicates that the object- 
slide is improperly centered. Usually this defect can only be corrected by 
an instrument-maker. If the difference is very small it may be ignored, 
but the adjustment should then be made on a point which is at about the 
mean distance for usual practice — say 150 feet. 

If the whole image appears to shift as the telescope is rotated, it indi- 
cates that the eyepiece is improperly adjusted. This defect is likewise 
usually corrected only by the maker. It does not interfere with instru- 
mental accuracy, but it usually causes the intersection of the cross-wires 
to be eccentric with the field of view. 

2. To make the axis of the level-tube parallel to the line of colli- 
mation. Raise the clips as far as possible. Swing the level 
so that it is parallel to a pair of opposite leveling-screws and 
clamp it. Bring the bubble to the middle of the tube by means 
of the leveling-screws. Take the telescope out of the wyes and 
replace it end for end, using extreme care that the wyes are not 
jarred by the action. If the bubble does not come to the center, 
correct one-half of the error by the vertical adjusting-screws at 
one end of the bubble. Correct the other half by the leveling- 
screws. Test the work by again changing the telescope end for 
end in the wyes. 

Care should be taken while making this adjustment to see 



APPENDIX. 561 

that the level-tube is vertically under the telescope. With the 
bubble in the center of the tube, rotate the telescope in the wyes 
for a considerable angle each side of the vertical. If the first 
half of the adjustment has been made and the bubble moves, it 
shows that the axis of the wyes and the axis of the level-tube 
are not in the same vertical plane although both have been made 
horizontal. By moving one end of the level-tube sidewise by 
means of the horizontal screws at one end of the tube, the two 
axes may be brought into the same plane. As this adjustment 
is liable to disturb the other, both should be alternately tested 
until both requirements are compHed ^dth. 

By these methods the axis of the bubble is made parallel to 
the axis of the wyes; and as this has been made parallel to the 
lines of collimation by means of the previous adjustment, the 
axis of the bubble is therefore parallel to the line of collimation. 

3. To make the line of collimation perpendicular to the vertical 
axis. Level up so that the instrument is approximately level 
over both sets of leveling-screw^s. Then, after leveHng carefully 
over one pair of screws, revolve the telescope 180° If it is not 
level, adjust half of the error by means of the capstan-headed 
screw imder one of the wyes, and the other half by the leveling- 
screws. Reverse again as a test. 

When the first two adjustments have been accurately made, good level- 
ing may always be done by bringing the bubble to the center by means of 
the leveling-screws, at every sight if necessary ^ even if the third adjust- 
ment is not mada Of course this third adjustment should be made as a 
matter of convenience, so that the line of collimation may be always level 
no matter in what direction it may be pointed, but it is not necessary to 
stop work to make this adjustment every time it is found to be defective. 

ADJUSTMENTS OF THE DUMPY LEVEL. 

1. To make the axis of the level-tube perpendicular to the vertical 
axis. Level up so that the instrument is approximately level 
over both sets of leveling-screws. Then, after leveHng care- 
fully over one pair of screws, revolve the telescope 180°. If 
it is not level, adjust one-half of the error by means of the adjust- 
ing-screws at one end of the bubble, and the other half by 
means of the leveling-screws. Reverse again as a test. 

2. To make the line of collimation perpendicular to the vertical 
axis. The method of adjustment is identical mth that for 
the transit (No. 4, p. 505) except that the cross-wire must be 



562 RAILROAD CONSTRUCTION. 

adjusted to agree with, the level-bubble rather than vice versa, as 
is the case with the corresponding adjustment of the transit; 
i.e., with the level-bubble in the center, raise or lower the hori- 
zontal cross-wire until it points at the mark known to be on 
a level with the center of the instrument. 

If the instrument has been well made and has not been dis- 
torted by rough usage, the cross- wires will intersect at the 
center of the field of view when adjusted as described. If they 
do not, it indicates an error which ordinarily can only be cor- 
rected by an instrument-maker. The error may be due to any 
one of several causes, which are 

(a) faulty centering of object-slide; 

(b) faulty centering of eyepiece; 

(c) distortion of instrument so that the geometric axis of 
the telescope is not perpendicular to the vertical axis. If the 
error is only just perceptible, it will not probably cause any 
error in the work. 



f 



EXPLANATORY NOTE ON THE USE OF THE TABLES. 

The logarithms here given are ''five-place/' but the last 
figure sometimes has a special mark over it (e.gr., 6) Tvhich indi- 
cates that one-half a unit in the last place should be added. 
For example 



the value 
.69586 
.69586 



includes all values between 
.6958575000 + and .6958624999. 
.6958625000 + and .6958674999. 



The maximum error in any one value therefore does not 
exceed one-quarter of a fifth-place unit. 

When adding or subtracting such logarithms allow a half-imit 
for such a sign. For example 

.69586 .69586 .69586 

.10841 .10841 .10841 

.12947 .12947 .12947 



.93374 .93375 .93375 

All other logarithmic operations are performed as usual and 
are supposed to be understood by the student. 

563 









TABLE I.—RADII OF CURVES. 








Deg 


0° 


1° 


3° 


3° 


Deg 


Min 


Radius. 


logH 


Radius, 


logM 


Radius. 


log It 


Radius. 


LogM 


iVIin 





00 


00 


5729.6 


3-75813 


2864-9 


3-45711 


1910.1 


3-28105 





1 


343775 


5-53627 


5635.7 


.75095 


2841-3 


.45351 


1899.5 


-27864 


1 


2 


171887 


5-23524 


5544.8 


.74389 


2818-0 


.44993 


1889.1 


•27625 


2 


3 


114592 


5.05915 


5456.8 


.73694 


2795-1 


.44639 


1878.8 


•27387 


3 


4 


85944 


4.93421 


5371.6 


.73010 


2772.5 


.44287 


1868.6 


.27151 


4 


5 


68755 


4-83730 


5288-9 


.72336 


2750.4 


.43939 


1858.5 


.26915 


__5 
6 


6 


57296 


4.75812 


5208-8 


3.71673 


2728.5 


3-43593 


1848.5 


3.26681 


7 


49111 


.69117 


5131-0 


.71020 


2707.0 


.43249 


1838-6 


.26448 


7 


8 


42972 


.63318 


5055-6 


.70377 


2685.9 


•42909 


1828-8 


.26217 


8 


9 


38197 


.58203 


4982-3 


.69743 


2665.1 


•42571 


1819-1 


.25986 


9 


10 


34377 


.53627 


4911-2 


.69118 


2644.6 


.42235 


1809-6 


-25757 


-19 
11 


11 


31252 


4.49488 


4842-0 


3.68502 


2624.4 


3.41903 


1800-1 


3-25529 


12 


28648 


.45709 


4774.7 


.67895 


2604.5 


.41572 


1790-7 


.25303 


12 


13 


26444 


.42233 


4709.3 


.67296 


2584.9 


.41245 


1781-5 


.25077 


13 


14 


24555 


.39014 


4645.7 


.66705 


2565.6 


.40919 


1772-3 


.24853 


14 


15 


22918 


.36018 


4583.8 


.66122 


2546.6 


.40597 


1763-2 


.24629 
3.24407 


15 
16 


16 


21486 
20222 
19099 


4.33215 


4523.4 


3.65547 


2527.9 


3-40276 


1754-2 


17 


.30582 


4464.7 


.64979 


2509.5 


.39958 


1745-3 


.24186 


17 


18 


.28100 


4407.5 


.64419 


2491-3 


•39642 


1736-5 


.23967 


18 


19 


18093 


.25752 


4351.7 


.63865 


2473-4 


•39329 


1727-8 


.23748 


19 


20 


17189 


.23524 


4297.3 


.63319 


2455-7 


.39017 


1719-1 


.23530 


20 
21 


21 


16370 


4.21405 


4244.2 


3.62780 


2438- 3 


3-38708 


1710-6 


3.23314 


22 


15626 


.19385 


4J92.5 


.62247 


2421-1 


-38401 


1702-1 


.23098 


22 


23 


14947 


.17454 


4142.0 


.61720 


2404-2 


.38097 


1693-7 


.22884 


23 


24 


14324 


.15606 


4092-7 


.61200 


2387-5 


.37794 


1685-4 


.22670 


24 


25 


13751 


.13833 


4044-5 


.60686 


2371-0 


-37494 


1677-2 


.22458 


25 
26 


26 


13222 


4.12130 


3997-5 


3.60178 


2354-8 


3-37195 


1669-1 


3.22247 


27 


12732 


.10491 


3951-5 


.59676 


2338-8 


.36899 


1661-0 


.22037 


27 


28 


12278 


.0891 


3906-6 


.59180 


2323-0 


.36604 


1653-0 


•21827 


28 


29 


11854 


.07387 


3862-7 


.58689 


2307-4 


•36312 


1645-1 


•21619 


29 


30 


11459 


.05915 


3819-8 


.58204 


2292-0 


.36021 


1637-3 


•21412 


-1? 
31 


31 


11090 


4.04491 


3777-9 


3.57724 


2276-8 


3.35733 


1629-5 


3-21206 


32 


10743 


.03112 


3736-8 


.57250 


2261-9 


•35446 


1621-8 


.21000 


32 


33 


10417 


.01776 


3696-6 


.56780 


2247-1 


•35162 


1614-2 


.20796 


33 


34 


10111 


4.00479 


3657-3 


•56316 


2232-5 


•34879 


1606-7 


.20593 


34 


35 


9822.2 


3.99221 


3618-8 


.55856 


2218-1 


.34598 


1599-2 


.20390 


35 
36 


36 


9549.3 


3.97997 


3581-1 


3.55401 


2203-9 


3.34318 


1591-8 


3.20189 


37 


9291-3 


.96807 


3544-2 


.54951 


2189-8 


•34041 


1584-5 


.19988 


37 


38 


9046.7 


•95649 


3508-0 


.54506 


2176-0 


•33765 


1577-2 


.19789 


38 


39 


8814.8 


.94521 


3472-6 


•54065 


2162-3 


•33491 


1570-0 


.1959S 


39 


40 


8594.4 


.93421 


3437-9 


.53629 


2148-8 


.33219 
3.32949 


1562-9 


.19392 


40 
41 


41 


8384.8 


3.92349 


3403-8 


3-53197 


2135-4 


1555-8 


3-19195 


42 


8185.2 


.91302 


3370-5 


.52769 


2122-3 


.32680 


1548-8 


.18999 


42 


43 


7994.8 


.90281 


3337-7 


•52345 


2109-2 


.32412 


1541-9 


.18804 


43 


44 


7813.1 


.89282 


3305.7 


•51925 


2096-4 


.32147 


1535-0 


.18610 


44 


45 


7639-5 


.88306 


3274.2 


.51510 


2083- 7 


.31883 


1528-2 


-18417 


49 
46 


46 


7473.4 


3.87352 


3243-3 


3-51098 


2071.1 


3-31621 


1521-4 


3-18224 


47 


7314.4 


.86418 


3213-0 


.50691 


2058.7 


.31360 


1514-7 


.18032 


47 


48 


7162.0 


.85503 


3183-2 


.50287 


2046-5 


.31101 


1508-1 


.17842 


48 


49 


7015.9 


.84608 


3154-0 


.49883 


2034-4 


.30843 


1501-5 


.17652 


49 


50 


6875. 6 


.83731 


3125.4 


.-49490 


2022-4 


.30587 
3-30332 


1495-0 
1488-5 


.17462 


50 
51 


51 


6740-7 


3.82871 


3097.2 


3-49097 


2010-6 


3.17274 


52 


6611-1 


.82027 


3069.8 


.48707 


1998-9 


-30079 


1482-1 


.17087 


52 


53 


6486-4 


.81200 


3042.4 


.48321 


1987-3 


.29827 


1475-7 


.16900 


53 


54 


6366-3 


.80388 


3015.7 


.47939 


1975-9 


.29577 


1469-4 


.16714 


54 


55 


6250-5 


.79591 


2989.5 


.47559 


1964-6 


.29328 


14.63-2 


.16529 


55 
56 


56 


6138-9 


3.78809 


2963.7 


3.47183 


1953.5 


3.29081 


1457-0 


3-16344 


57 


6031-2 


.7804(5 


2938-4 


.46811 


1942-4 


.28835 


1450-8 


.16161 


57 


58 


5927-2 


.77285 


2913-5 


.46441 


1931-5 


.28590 


1444-7 


.15978 


58 


59 


5826.8 


.76542 


2889. 


.46075 


1920-7 


.28347 


1438-7 


.15796 


59 


60 


5729-6 


.75813 


2864-9 


.45711 


1910-1 


.28105 


1432-7 


.15615 


60 










50 


4 








; 











TABLE 


I.— RADII OF CURVES. 








Deg 

'Min 


4° 


5° 


6° 


r 


Deg 


Radius. 


log It 


Radius. 


log It 


Radius. 


logn 


Radius. 


logH 


Min 




1 
2 

I 

6 


1432-7 
1426-7 
1420-8 
1415-0 
1409-2 
1403-5 

1397-8 
1392-1 
1386-5 
1380-9 
1375-4 


3 


-15615 
•15434 
.15255 
.15076 
.14897 
-14720 


1146.3 
1142-5 
1138-7 
1134-9 
1131-2 
1127-5 


3-05929 
•05784 
-05640 
-05497 
•05354 
-05211 

3-05069 
•04928 
.04787 
.04646 
-04506 


955 
952 
950 
947 
944 
942 


37 
72 
09 
48 
88 
29 


2-98017 
-97896 
•97776 
•97657 
•97537 
-97418 


819-02 
817-08 
815-14 
813-22 
811-30 
809-40 


2.91329 
.91226 
.91123 
.91021 
.90918 
.90816 



1 
2 
3 
4 
5 


3 


-14543 
.14367 
.14191 
.14017 
.13843 


1123-8 
1120-2 
1116-5 
1112-9 
1109-3 


939 
937 
934 
932 
929 


72 
16 
62 
09 
57 


2-97300 
•97181 
-97063 
.96945 
-96828 


807-50 
805-61 
803-73 
801-86 
800-00 


2-90714 
.90612 
.90511 
.90410 
-90309 


6 
7 
8 
9 
10 


11 

*13 
^4 
(15__ 

.'16 
(17 

18 

19 

20 

21 

22 

23 

24 
'25 
(^ 

27 
'28 

29 

30 


1369-9 
1364-5 
1359-1 
1353-8 
1348-4 


3 


-13669 
.13497 
.13325 
•13154 
12983 


1105-8 
1102-2 
1098-7 
1095-2 
1091-7 


3^04366 
•04227 
.04088 
•03949 
^03811 


927 
924 
922 
919 
917 


07 
58 
10 
64 
19 


2-96711 
•96594 
•96478 
•96361 
•96246 


798-14 
796-30 
794-46 
792-63 
790-81 


2-90208 
.90107 
.90007 
.89907 
.89807 


11 
12 
13 
14 
15 


1343-2 
1338-0 
1332-8 
1327-6 
1322-5 


3 


12813 
12644 

■12475 
12307 

-12140 


1088-3 
1084-8 
1081-4 
1078-1 
1074-7 


3-03674 
•03537 
.03400 
.03264 
•03128 


914 
912 
909 
907 
905 


75 
33 

92 
52 
13 


2-96130 
-96015 
•95900 
•95785 
-95671 


789-00 
787-20 
785-41 
783-62 
781-84 


2.89708 
.89608 
•89509 
•89410 
•89312 


16 
17 
18 
19 
20 


1317-5 
1312-4 
1307-4 
1302-5 
1297-6 


3 


11974 
11808 
11642 
11477 
11313 


1071-3 
1068-0 
1064-7 
1061^4 
1058^2 


3^02992 
^02857 
.02723 
.02589 
•02455 


902 
900 
898 
895 
893 


76 
40 
05 
71 
39 


2-95557 
•95443 
•95330 
•95217 
•95104 


780-07 
778-31 
776-55 
774-81 
773-07 


2-89213 
•89115 
•89017 
•88919 
•88821 


21 

22 
23 
24 
25 


1292.7 
1287.9 
1283.1 
1278.3 
1273-6 


3 


11150 
10987 
10825 
10663 
10502 


1054-9 
1051-7 
1048.5 
1045-3 
1042-1 


3-02322 
•02189 
.02056 
•01924 
-01792 


891 
888 
886 
884 
881 


08 
78 
49 

21 
95 


2-94991 
-94879 
•94767 
•94655 
-94544 


771-34 
769-61 
767-90 
766-19 
764-49 


2-88724 
•88627 
.88530 
•88433 
•88337 


26 
27 
28 
29 
30 


31 
32 
133 
34 
35 


1268-9 
1264-2 
1259-6 
1255-0 
1250-4 


3 


10341 
10182 
10022 
09864 
09705 


1039-0 
1035-9 
1032-8 
1029.7 
1026-6 


3-01661 
•01530 
.01400 
.01270 
•01140 


879 
877 
875 
873 
870 


69 
45 
22 
00 
80 


2-94433 
-94322 
-94212 
.94101 
-93991 


762-80 
761-11 
759-43 
757-76 
756-10 


2 •88241 
•88145 
.88049 
.87953 
•87858 


31 
32 
33 
34 
35 


36 
37 
38 
39 
40 


1245-9 
1241-4 
1236-9 
1232-5 
1228-1 


3 


09548 
09391 
09234 
09079 
08923 


1023-5 
1020-5 
1017-5 
1014^5 
1011.5 


3^01010 
.00882 
.00753 
.00625 
.00497 


868 
866 
864 
862 
859 


60 
41 
24 
07 
92 


2.93882 
•93772 
•93663 
.93554 
.93446 


754-44 
752-80 
751-16 
749-52 
747-89 


2 •87762 
•87668 
.87573 
.87478 
.87384 


36 
37 
38 
39 
40 


41 
42 
43 
44 
45 


1223-7 
1219-4 
1215-1 
1210-8 
1206-6 


3 


08769 
08614 
08461 
08308 
08155 


1008.6 
1005-6 
1002-7 
999-76 
996-87 


3 • 00370 

. 00242 

3-00116 

2-99989 

.99863 


857 
855 
853 
851 
849 


78 
65 
53 
42 
32 


2.93337 
•93229 
•93122 
•93014 
•92907 


746-27 
744-66 
743-06 
741-46 
739-86 


2.87290 
.87196 
.87102 
.87008 
.86915 


41 

42 
43 
44 
45 


46 
47 
48 
49 
50 


1202-4 
1198-2 
1194-0 
1189-9 
1185-8 


3- 


08003 
07852 
07701 
07550 
07400 


993-99 
991.13 
988-28 
985-45 
982-64 


2-99738 
.99613 
. 99488 
. 99363 
- 99239 


847 
845 
843 
841 
838 


23 
15 
08 
02 
97 


2.92800 
•92693 
•92587 
•92480 
•92374 


738-28 
736-70 
735-13 
733-56 
732-01 


2.86822 
.86729 
.86636 
.86544 
•86451 


46 
47 
48 
49 
50 


51 
52 
53 
54 
55 


1181-7 
1177.7 
1173-6 
1169-7 
1165-7 


3- 


07251 
07102 
06954 
06806 
06658 


979-84 
977-06 
974-29 
971-54 
968-81 


2. 99115 
.98992 
.S?8869 
.9 8746 
.9 8624 

2.98501 
-98380 
-98258 
-98137 
^98017 


836 
834 
832 
830 
828 


93 
90 
89 
88 
88 


2.92269 
•92163 
.92058 
.91953 
•91849 


730-45 
728-91 
727-37 
725-84 
724-31 


2-86359 
.86267 
.86175 
.86084 
.85992 


51 
52 
53 
54 
55 


56 
57 
58 
59 
60 


1161-8 
1157-9 
1154-0 
1150-1 
1146.3 


3- 


06511 
06365 
06219 
06074 
05929 


966-09 
963-39 
960-70 
958-03 
955-37 


826 
824 
822 
820 
819 


89 
91 
93 
97 
02 


2-91744 

.9164(5 

I .91536 

.91433 

.91329 


722.79 
721.28 
719-77 
718-27 
716.78 


2-85901 
.85810 
.85719 
.85629 
.85538 


56 
57 
58 
59 
60 



565: 









TABLE I.— RADII OF CURVES. 








Deg. 


8^ 


9^ 


10° 


11« 


Deg 


Min. 


Radius. 


Log-B 


Radius. 


Log 12 


Radius. 


Log 12 


Radius. 


Log 12 


Min 



1 
2 
3 

4 
5 


716.78 
715.29 
713.81 
712.34 
710.87 
709.40 


2.85538 
. 85448 
.85358 
.85268 
.85178 
.85089 


637.27 
636.10 
634.93 
633.76 
632.60 
631.44 


2.80432 
•80352 
•80272 
•80192 
•80113 
•80033 


573.69 
572-73 
571-78 
570-84 
569-90 
568-96 


2-75867 
•75795 
.75723 
.75651 
.75579 
•75508 


521-67 
520-88 
520-10 
519-32 
518-54 
517-76 


2-71739 
.71674 
.71608 
.71543 
.71478 
.71413 



1 
2 
3 
4 
5 


6 
7 
8 
9 
10 


707.95 
706.49 
705.05 
703.61 
702.17 


2.85000 
.84911 
.84822 
.84733 
•84644 


630.29 
629.14 
627.99 
626.85 
625.71 


2^79954 
•79874 
•79795 
.79716 
•79637 


568.02 
567.09 
566.16 
565.23 
564-31 


2.75436 
.75365 
.75293 
.75222 
.75151 


516-99 
516-21 
515^44 
514^68 
513-91 


2-71348 
.71283 
.71218 
.71153 
-71088 


6 
7 
8 
9 
10 


11 
12 
13 
14 
15 


700.75 
699.33 
697.91 
696.50 
695.09 


2.84556 
.84468 
.84380 
.84292 
.84205 


624.58 
623.45 
622.32 
621.20 
620.09 
618.97 
617.87 
616.76 
615.66 
614.56' 


2.79558 
.79480 
.79401 
•79323 
•79245 


563-38 
562^47 
561^55 
560.64 
559.73 


2.75080 
.75009 
.74939 
.74868 
•74798 


513.15 
512.38 
511.63 
510 • 87 
510^11 


2-71024 
.70959 
.70895 
.70831 
-70767 


11 
12 
13 
14 
15 


16 
17 
18 
19 
20 


693.70 
692.30 
690.91 
689.53 
688.16 


2.84117 
.84029 
.83942 
.83855 
•83768 


2^79167 
•79089 
.79011 
.78934 
•78856 


558.82 
557-92 
557-02 
556-12 
555-23 


2.74727 
.74657 
.74587 
.74517 
.74447 


509^36 
508 • 61 
507-86 
507-12 
506-38 


2-70702 
.70638 
.70575 
-70511 
. 70447 


16 
17 
18 
19 
20 


21 
22 
23 
24 
25 


686.78 
685.42 
684.06 
682.70 
681.35 


2.83682 
.83595 
.83509 
.83423 
.83337 


613.47 
612.38 
611.30 
610.21 
609.14 


2 •78779 
.78702 
.78625 
.78548 
.78471 


554-34 
553-45 
552-56 
551-68 
550-80 


2.74377 
.74307 
.74238 
.74168 
•74099 


505-64 
504-90 
504-16 
503-42 
502-69 


2.70383 
.70320 
.70257 
.70193 
.70130 


21 
22 
23 
24 
25 


26 
27 
28 
29 
30 


680.01 
678.67 
677.34 
676.01 
674.69 


2.83251 
.83166 
.83080 
.82995 
•82910 


608.06 
606.99 
605.93 
604.86 
603.80 


2.78395 
.78318 
.78242 
.78165 
•78089 


549-92 
549-05 
548 • 17 
547^30 
546.44 


2.74030 
.73961 
.73892 
.73823 
-73754 


501-96 
501-23 
500-51 
499-78 
499.06 


2.70067 
.70004 
•69941 
.69878 
•69815 


26 
27 
28 
29 
30 


31 
32 
33 
34 
35 


673.37 
672.06 
670.75 
669.45 
668.15 


2.82825 
•82740 
.82656 
.82571 
•82487 


602.75 
601.70 
600.65 
599-61 
598.57 


2.78013 
.77938 
.77862 
.77786 
.77711 


545-57 
544-71 
543-86 
543-00 
542-15 


2.73685 
.73617 
.73548 
-73480 
-73412 


498-34 
497-62 
496-91 
496-19 
495-48 


2.69752 
.69690 
.69627 
.69565 
.69503 


31 
32 
33 
34 
35 


36 
37 
38 
39 
40 


666.86 
665.57 
664.29 
663.01 
661.74 


2 •82403 
.82319 
.82235 
.82152 
•82068 


597.53 
596.50 
595.47 
594.44 
593.42 
592.40 
591.38 
590.37 
589.36 
588. 36 


2-77636 
.77561 
•77486 
.77411 
•77336 


541-30 
540-45 
539-61 
538.76 
537-92 


2-73343 
.73275 
.73207 
.73140 
-73072 


494-77 
494-07 
493-36 
492-66 
491-96 


2.69440 
•69378 
•69316 
•69255 
.69192 


36 
37 
38 
39 
40 


41 
42 
43 
44 

45 


660.47 
659.21 
657.95 
656.69 
655.45 


2.81985 
.81902 
•81819 
.81736 
.81653 

2.81571 
•81489 
•81406 
.81324 
.81243 


2 •77261 
•77187 
•77112 
•77038 
•76964 


537.09 
536-25 
535.42 
534.69 
533-77 


2 . 73004 
.72937 
.72869 
.72802 
.72735 


491.26 
490.56 
489.86 
489.17 
488. 48 


2.69131 
.69069 
•69007 
•68946 
•68884 


41 
42 
43 

44 
45 


46 
47 
48 
49 
50 


654.20 
652.96 
651.73 
650.50 
649.27 


587.36 
586. 36 
585.36 
584.37 
583. 38 


2^76890 
•76816 
•76742 
.76669 
•76595 


532-94 
532-12 
531-30 
530-49 
529-67 


2.72668 

•72601 

•72534 

.72467 

72401 


487.79 
487.10 
486.42 
485.73 
485 05 


2.68823 
.68762 
.68701 
.68640 
.68579 


46 
47 
48 
49 
50 


51 
52 
53 
54 
65 


648.05 
646.84 
645.63 
644.42 
643.22 


2.81161 
•81079 
.80998 
•80917 
.80836 


582.40 
581.42 
580.44 
579.47 
578.49 


2-76522 
.76449 
.76376 
•76303 
.76230 


528-86 
528-05 
527-25 
526-44 
525-64 
524.84 
524-05 
523.25 
522.46 
S21-67 


2.72334 
.72267 
.72201 
-72135 
-72069 


484.37 
483.69 
483. 02 
482.34 
481.67 


2.68518 
.68457 
•68396 
•68335 
.68275 


51 
52 
53 
54 
55 


56 
57 
58 
^9 
(JO 


642.02 
640.83 
639.64 
638.45 
637.27 


2.80755 
.80674 
.80593 
.80513 
.80432 


577.53 
576.56 
575.60 
574.64 
573.69 


2.76157 
•76084 
•76012 
•75939 
•75867 


2-72003 
-71937 
-71871 
-71805 
-71739 


481.00 
480.33 
479.67 
479-00 
478-34 


2.68214 
•68154 
•68094 
•68033 
.67973 


56 
57 
58 
59 
60 



566: 













TABLE I. 


—RADII OF CURVES. 










Deg. 


Radius. 


logM 


Deg. 


Radius. 


logH 


Deg. 

16° 

5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

17° 

5 
10 
15 
20 
_2_5 
30 
35 
40 
45 
50 
55 

18° 
5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 

19° 

5 
10 
15 
20 
25 


Radius. 


Log^K 


Deg. 

21° 

10 
20 
30 
40 
50 


Radius 


Log-B 


12° 

2 
4 
6 
8 


478 
477 
475 
474 
473 


34 
02 
71 
40 
10 


2 


67973 
67853 
67734 
67614 
67495 


14° 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 


410 
409 
408 
407 
406 


28 
31 
34 
38 
42 


2 


.61307 
61205 
61102 
61000 
60898 


359 
357 
355 
353 
351 
350 


.26 
42 
59 
77 
98 
21 


2 


.55541 
.55317 
.55094 
.54872 
.54652 
54432 


274 
272 
270 
268 
266 
264 


37 
.23 
• 13 
06 
02 
02 


2-43833 
.43494 
.43157 
.42823 
.42492 


10 


471 
470 
469 
467 
466 


81 
53 
25 
98 
72 


2 


67376 
67258 
67140 
67022 
66905 


405 
404 
403 
402 
401 


47 
53 
58 
65 
71 


2 


60796 
60694 
60593 
60492 
60391 


•42163 


12 
14 
16 
18 


348 
346 
344 
343 
341 
339 


45 
71 
99 
29 
60 
93 


2 


54214 
53997 
53780 
53565 
53351 
53138 


22° 
10 
20 
30 
40 
50 


262 
260 
258 
256 
254 
252 


.04 
10 
18 
29 
43 
60 


2.41837 
.41513 
.41192 
•40873 


20 
22 


465 
464 
462 
461 
460 


46 
21 
97 
73 
50 


2 


66788 
66671 
66555 
66439 
66323 


400 
399 
398 
398 
397 


78 
86 

94 
02 
11 


2 


60291 
60190 
60090 
59990 
59891 


.40557 
.40243 


24 
26 
28 


338 
336 
335 
333 
331 
330 

328 
327 
325 
324 
322 
321 


27 
64 
01 
41 
82 
24 
68 
13 
60 
09 
59 
10 


2 


52927 
52716 
52506 
52297 
52090 
51883 


23° 

10 
20 
30 
40 
50 

24° 
10 
20 
30 
40 
50 

25° 

30 
26° 

30 

27° 

30 

28° 

30 

29^ 

30 
30° 

30 

31° 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

52 

54 

56 

58 

60 


250 
249 
247 
245 
243 
242 


79 
01 
26 
53 
82 
14 


2.39931 
39622 
.39315 


30 
32 
34 


459 
458 
456 
455 

454 

453 
452 
450 
449 
448 


28 
06 
85 
65 
45 
26 
07 
89 
72 
56 


2 


66207 
66092 
65977 
65863 
65748 


30 
32 
34 
36 
38 
40 
42 
44 
46 
48 
50 
52 
54 
56 
58 

15° 

2 

4 

6 

8 

10 

12 

14 

16 

18 


396 
395 
394 
393 
392 

391 
390 
389 
389 
388 


20 
30 
40 
50 
61 
72 
84 
96 
08 
21 


2 


59791 
59692 
59593 
59494 
59396 


.39015 
38707 
38407 


36 
38 


2 


51677 
51472 
51269 
51066 
50864 
50663 


240 
238 
237 
235 
234 
232. 


49 
85 
24 
65 
08 
54 


2.38109 
.37813 


40 
42 
44 
46 


2 


65634 
65521 
65407 
65294 
65181 


2 


59298 
59199 
59102 
59004 
58907 


.37519 
•37227 
•36937 
.36649 


48 


319 
318 

316 
315 
313 

312 

311 
309 
308 
306 
305 
304 
302 
301 
300 
299 
297 
296 


62 
16 
71 
28 
86 
45 
06 
67 
30 
95 
60 
27 
94 
63 
33 
04 
77 
50 


2 


50464 
50265 
50067 
49869 
49673 
49478 


231. 
226. 
222. 
218 


01 
55 
27 
15 


2.36363 


50 
52 


447 
446 
445 
443 
442 


40 
24 
09 
95 
81 


2 


65089 
64957 
64845 
64733 
64622 


387 
386 
385 
384 
383 


34 
48 
62 
77 
91 


2 


58809 
58713 
58616 
58519 
58423 


.35517 
•34688 
.33875 


56 
S8 


214. 
210- 
206. 
203. 


18 
36 
68 
13 


2-33078 
.32296 


13° 

9 


441 

440 

439 

438 

437^ 

436 

435 

433 

432 

431 


68 
56 
44 
33 
22 
12 
02 
93 
84 
76 


2 


64511 
64400 
64290 
64180 
64070 


383 
382 
381 
380 
379 


06 
22 
38 
54 
71 


2 


58327 
58231 
58135 
58040 
57945 


2 


49284 
49090 
48898 
48706 
48515 
48325 


•31529 
•30776 


4 
6 
8 


199. 
196. 
193. 
190. 


70 
38 
19 
09 
10 
40 
05 
02 
28 
80 
58 
58 
79 
19 


2.30037 
•29310 
.28597 
•2789§ 


10 
12 
14 
16 
18 


2 


63960 
63851 
63742 
63633 
63524 


378 
378 
377 
376 
375 


88 

05 
23 
41 
60 


2 


57850 
57755 
57661 
57566 
57472 


2 


48136 
47948 
4776C 
47573 
47388 
47203 


187 

181 

176. 

171 

166 

161 

157 

153 

149 

146 


2.27207 
•25863 
•24563 
•23303 


^0 


430 
429 
428 
427 
426 


69 
62 
56 
50 
44 


2 


63416 
63308 
63201 
63093 
62986 


20 
22 
24 
26 
28 
30 
32 
34 
36 
38 


374 
373 
373 
372 
371 


79 
98 
17 
37 
57 


2 


57378 
57284 
57191 
57097 
57004 


.22083 


22 
24 
26 
28 


2^20899 
•19749 
.18633 
.17547 


30 
35 
40 
45 
50 
55 

30° 

5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 
21° 


295 
294 
292 
291 
290 
289 


25 
00 
77 
55 
33 
13 


2 


47018 
46835 
46652 
46471 
46289 
46109 


30 


425 
424 
423 
422 
421 


40 
35 
32 
28 
26 


2 


62879 
62773 
62666 
62560 
62454 


370 
369 
369 
368 
367 


78 
99 
20 
42 
64 


2 


56911 
56819 
56726 
56634 
56542 


•16492 


32 
34 
36 
38 


142 
139 
136 
133 
130 


77 
52 
43 
47 
66 


2 •15465 
• 14464 
•1348| 
•12539 


287 
286 
285 
284 
283 
282 


94 
76 
58 
42 
27 
12 


2 


45930 
45751 
45573 
45396 
45219 
45044 


40 


420 
419 
418 
417 
416 
415 
414 
413 
412 
411 


23 
22 
20 
19 
19 
19 
20 
21 
23 
25 


2 


62349 
62243 
62138 
62034 
61929 


40 
42 
44 
46 
48 
50 
52 
54 
56 
58 
16° 


366 
366 
365 
364 
363 


86 
09 
31 
55 
78 


2 


56450 
56358 
56266 
.56175 
56084 


.11613 


42 
44 
46 
48 


127 
125 
122 
120 
118 


97 
39 
93 
57 

31 


2.10709 
.09827 
.08965 
.08125 


280 
279 
278 
277 
276 
275 


99 
86 
75 
64 
54 
45 


2 


44869 
44694 
44521 
44348 
44176 
44004 


50 


2 


61825 
61721 
61617 
61514 
61410 


363 
362 
361 
360 
360 


02 
26 
51 
76 
01 


2 


.55993 
.55902 
.55812 
.55721 
.55631 


.07302 


52 
54 
56 
58 


114 
110 
106 
103 
100 


06 
13 
50 
13 
00 


2.05713 
.04192 
•02736 
•01340 


274 


37 


2 


43833 


14° 


410 


28 


2 


61307 


359 


26 


2 


•55541 


2.00000 



567 



TABLE H.—TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS 
FOR A 1° CURVE. j 



A 


T^ng- 


Ext. 
Dist. 


Long 
Chord 


A 


T^ff' 


Ext. 
Dist. 


Long 
Chord 


A 

21° 

10 
20 
30 
40 
50 

22° 
10 
20 
30 
40 
50 


T^g- 


Ext. 
Dst. 

JE. 


Long 
Chorci 

xc 


1° 

10' 
20 
30 
40 
50 


50.00 
58.34 
66.67 
75.01 
83.34 
91.68 


0.218 
0.297 
0.388 
0.491 
0.606 
0.733 


100 
116 
133 
150 
166 
183 


-00 
-67 
-33 
-00 
66 
33 


11° 

10 
20 
30 
40 
50 


551 
560 
568 
576 
585 
593 


.70 
.11 
.53 
.95 
36 
79 


26.500 
27.313 
28.137 
28.9-74 
29.824 
30.686 


1098 
1114 
1131 
1148 
1164 
1181 


-3 

■\ 

1 


1061-9 
1070-6 
1079.2 
1087-8 
1096-4 
1105-1 


97.58 
99-15 
100-75 
102.35 
103.97 
105-60 


2088-3 
2104.7 
2121.1 
2137.4 
2153.8 
2170.2 


2° 
10 
20 
30 
40 
50 


100.01 
108.35 
116.68 
125.02 
133.36 
141.70 


0.873 
1.024 
1.188 
1.364 
1.552 
1.752 


199 
216 
233 
249 
266 
283 


99 
66 
32 
98 
65 
31 


12° 

10 
20 
30 
40 
50 
13° 
10 
20 
30 
40 
50 

14° 

10 
20 
30 
40 
50 

15° 

10 
20 
30 
40 
50 

16° 

10 
20 
30 
40 
50 


602 
610 
619 
627 
635 
644 


• 21 
64 
07 

.50 
93 
37 


31.561 
32-447 
33-347 
34-259 
35-183 
36-120 


1197 
1214 
1231 
1247 
1264 
1280 


8 

4 


-5 
1 

.7 


1113-7 
1122-4 
1131-0 
1139-7 
1148-4 
1157-0 


107-24 
108-90 
110-57 
112-25 
113-95 
115-66 


2186.5 
2202.9 
2219.2 
2235.6 
2251.9 
2268.3 


3° 

10 
20 
30 
40 
50 


150.04 
158.38 
166-72 
175.06 
183.40 
191.74 


1.964 
2.188 
2.425 
2.674 
2.934 
3-207 


299 
316 
333 
349 
366 
383 


97 
63 
29 
95 
61 
27 


652 
661 
669 
678 
686 
695 


81 
25 
.70 
15 
60 
06 


37.069 
38.031 
39.006 
39.993 
40-992 
42-004 


1297 
1313 
1330 
1346 
1363 
1380 


.2 
-8 
-3 
-9 
4 
-0 


23° 

10 
20 
30 
40 
50 


1165-7 
1174-4 
1183.1 
1191.8 
1200-5 
1209-2 


117-38 
119-12 
120-87 
122-63 
124-41 
126-20 


2284-6 
2301.0 
2317.3 
2333.6 
2349.9 
2366.2 


4° 
10 
20 
30 
40 
50 


200.08 
208.43 
216.77 
225.12 
233.47 
241.81 


3-492 
3-790 
4.099 
4-421 
4.755 
5.100 


399 
416 
433 
449 
466 
483 


92 
58 
24 
89 
54 
20 


703 
711 
720 
728 
737 
745 


51 
97 
44 
90 
37 
85 


43-029 
44-066 
45.116 
46.178 
47.253 
48-341 


1396 
1413 
1429 
1446 
1462 
1479 


-5 
-1 
-6 
-2 
-7 
2 


24° 
10 
20 
30 
40 
50 

25° 
10 
20 
30 
40 
50 


1217-9 
1226-6 
1235-3 
1244-0 
1252-8 
1261-5 


128-00 
129-82 
131-65 
133-50 
135-36 
137-23 


2382.5 
2398.8 
2415.1 
2431.4 
2447.7 
2464.0 


5° 
10 
20 
30 
40 
50 


250.16 
258.51 
266.86 
275.21 
283.57 
291.92 


5.459 
5.829 
6.211 
6.606 
7.013 
7.432 

7.863 
8.307 
8.762 
9.230 
9.710 
10.202 


499 
516 
533 
549 
566 
583 


85 
50 
15 
80 
44 
09 


754 
762 
771 
779 
788 
796 


32 
80 
29 
77 
26 
75 


49-441 
50-554 
51-679 
52-818 
53-969 
55-132 


1495 
1512 
1528 
1545 
1561 
1578 


-7 
3 
8 
3 
8 
3 


1270-2 
1279-0 
1287-7 
1296-5 
1305-3 
1314-0 
1322-8 
1331-6 
1340-4 
1349-2 
1358-0 
1366-8 


139-11 
141-01 
142-93 
144-85 
146-79 
148-75 


2480.2 
2496.5 
2512.8 
2529.0 
2545.3 
2561.5 


6° 

10 
20 
30 
40 
50 


300.28 
308.64 
316.99 
325.35 
333.71 
342.08 


599 
616 
633 
649 
666 
682 


73 
38 
02 
66 
30 
94 


805 
813 
822 
830 
839 
847 


25 
75 
25 
76 
27 
78 


56.309 
57.498 
58.699 
59.914 
61.141 
62.381 
63.634 
64.900 
66.178 
67.470 
68.774 
70.091 


1594 
1611 
1627 
1644 
1660 
1677 


8 
3 
8 
3 
8 
3 


26° 

10 
20 
30 
40 
50 


150-71 
152-69 
154-69 
156-70 
158-72 
160-76 


2577.8 
2594.0 
2610.3 
2626.5 
2642.7 
2658.9 


r 

10 
20 
30 
40 
50 


350-44 
358-81 
367-17 
375.54 
383. 91 
392.28 


10.707 
11.224 
11.753 
12.294 
12.847 
13.413 


699 
716 
732 
749 
766 
782 


57 
21 
84 
47 
10 
73 


17° 

10 
20 
30 
40 
50 


856 
864 
873 
881 
890 
898 


30 
82 
35 
88 
41 
95 


1693 
1710 
1726 
1743 
1759 
1776 


8 
3 
8 

2 
7 

2 


27° 
10 
20 
30 
40 
50 


1375-6 
1384-4 
1393-2 
1402.0 
1410-9 
1419-7 


162-81 
164-87 
166-95 
169-04 
171-15 
173-27 


2675.1 
2691.3 
2707.5 
2723.7 
2739.9 
2756.1 


8° 
10 
20 
30 
40 
50 


400.66 
409.03 
417.41 
425.79 
434.17 
442.55 


13.991 
14.582 
15.184 
15.799 
16.426 
17.066 


799 
815 
832 
849 
865 
882 


36 
99 
61 
23 
85 
47 


18° 
10 
20 
30 
40 
50 

19° 

10 
20 
30 
40 
50 

20° 

10 
20 
30 
40 
50 

21° 


907 

916 

924 

933 

941- 

950 


49 
03 
58 
13 
69 
25 


71.421 
72.764 
74-119 
75-488 
76-869 
78-264 


1792 

1809 

1825 

1842- 

1858- 

1874- 


6 
1 
5 

4 
9 


28° 
10 
20 
30 
40 
50 


1428-6 
1437-4 
1446-3 
1455-1 
1464-0 
1472-9 


175-41 
177-55 
179-72 
181-89 
184-08 
186-29 


2772.3 
2788.4 
2804.6 
2820.7 
2836.9 
2853.0 


9° 

10 
20 
30 
40 
50 


450.93 
459.32 
467.71 
476.10 
484.49 
492.88 


17.717 
18.381 
19.058 
19.746 
20.447 
21.161 


899 
915 
932 
948 
965 
982 


09 
70 
31 
92 
53 
14 


958 
967 
975 
984 
993 
1001 


81 
38 
96 
53 
12 
70 


79-671 
81-092 
82-525 
83.972 
85.431 
86-904 


1891- 
1907- 
1924- 
1940 
1957 
1973 


3 
8 

2 
6 
1 
5 


29° 

10 
20 
30 
40 
50 

30° 

10 
20 
30 
40 
50 


1481-8 
1490-7 
1499-6 
1508-5 
1517-4 
1526-3 


188-51 
190-74 
192-99 
195-25 
197-53 
199-82 


2869-2 
2885-3 
2901-4 
2917-6 
2933-7 
2949-8 


10° 

10 
20 
30 
40 
50 


501.28 
509.68 
518-08 
526.48 
534-89 
543.29 


21.886 
22-624 
23-375 
24-138 
24-913 
25-700 


998 
1015 
1031 
1048 
1065 
1081 


74 
35 
95 
54 
14 
.73 


1010 
1018 
1027 
1036 
1044 
1053 


29 
89 
49 
09 
70 
31 


88-389 
89-888 
91-399 
92-924 
94-462 
96-013 


1989 
2006 
2022 
2039 
2055 
2071 


9 
3 
7 
1 
5 
9 


1535-3 
1544-2 
1553-1 
1562-1 
1571-0 
1580-0 


202-12 
204-44 
206-77 
209-12 
211-48 
213-86 


2965-9 
2982.0 
2998.1 
3014.2 
3030-2 
3046.3 


11° 


551. 70126. 500 


1098 


^ 


1061 


_93. 


97-577 


2088- 


A 


31° 


1589.0 


216-25 


3062.4 



568 



TABLE II.— TANGENTS. 



EXTERNAL DISTANCES, 
FOR A 1° CURVE. 



AND LONG CHORDS 



A 


Tang 


■ 


Ext. 
Dist. 


Long 
Chord 


A 


T^g' 


.Ex*' 
Dist. 


Long 
Chord 
JL€. 


A 


T^g 


•• 


Ext. 
Dist. 


Long 
Chord 

xc. 


31° 

10' 
20 
30 
40 
50 


1589 
1598 
1606 
1615 
1624 
1633 
1643 
1652 
1661 
1670 
1679 
1688 




9 
9 
9 
9 




1 
1 


216 
218 
221 
223 
225 
228 


25 
66 
08 
51 
96 
42 


3062 
3078 
3094 
3110 
3126 
3142 


4 
4 
5 
5 
6 
6 


41° 

10 
20 
30 
40 
50 

43° 
10 
20 
30 
40 
50 

43° 

10 
20 
30 
40 
50 


2142.2 
2151.7 
2161.2 
2170.8 
2180-3 
2189-9 


387 
390 
394 
397 
400 
404 


384013 
71 4028 
06 4044 
43 4059 
82 4075 
22 4091 


-1 
-7 
-3 
.9 
.5 
1 


51° 

10 
20 
30 
40 
50 

53° 
10 
20 
30 
40 
50 

53° 

10 
20 
30 
40 
50 


2732 
2743 
2753 
2763 
2773 
2784 


9 
1 
4 

9^ 

2 


618 

622 
627 
631 
636 
640 


39 
81 
24 
69 
16 
66 


4933 
4948 
4963 
4978 
4993 
5008 


4 
4 
4 
4 
4 
4 


33° 

10 
20 
30 
40 
50 


230 
233 
235 
238 
240 
243 
246 
248 
251 
253 
256 
259 


90 
39 
90 
43 
96 
52 

08 
66 
26 
87 
50 
14 


3158 
3174 
3190 
3206 
3222 
3238 


6 
6 
6 
6 
6 
6 


2199-4 
2209-0 
2218.6 
2228.1 
2237-7 
2247-3 


407 
411 
414 
417 
421 
424 


64 
07 
52 
99 
48 
98 


4106 
4122 
4137 
4153 
4168 
4184 


6 
2 
7 
3 
8 
3 


2794 
2804 
2815 
2825 
2835 
2846 


5 
9 
2 
6 
9 
3 


645 
649 
654 
658 
663 
668 


17 
70 
25 
83 
42 
03 


5023 
5038 
5053 
5068 
5083 
5098 


4 
4 
4 
3 
3 
2 


33° 

10 
20 
30 
40 
50 


1697 
1706 
1715 
1724 
1733 
1742 


2 
3 
3 

4 
5 
6 


3254 
3270 
3286 
3302 
3318 
3334 


6 
6 
6 
5 
5 
4 


2257-0 
2266-6 
2276-2 
2285-9 
2295-6 
2305-2 


428 
432 
435 
439 
442 
446 


50 
04 
59 
16 
75 
35 


4199 
4215 
4230 
4246 
4261 
4277 


8 
3 
8 
3 
8 
3 


2856 
2867 
2877 
2888 
2898 
2908 


7 
1 
5 

I 

9 


672 
677 
681 
686 
691 
696 


66 
32 
99 
68 
40 
13 


5113 
5128 
5142 
5157 
5172 
5187 


1 

9 
8 
7 
6 


34° 

10 
20 
30 
40 
50 


1751 
1760 
1770 
1779 
1788 
1797 


7 
8 



1 
2 
4 


261 
264 
267 
269 
272 
275 


80 
47 
16 
86 
58 
31 


3350 
3366 
3382 
3398 
3414 
3430 


4 
3 

2 
2 
1 



44° 

10 
20 
30 
40 
50 

45° 

10 
20 
30 
40 
50 


2314-9 
2324-6 
2334-3 
2344-1 
2353-8 
2363-5 


449 
453 
457 
460 
464 
468 


98 

62 
27 
95 
64 
35 


4292 
4308 
4323 
4339 
4354 
4369 


7 
2 
6 

5 
9 


54° 
10 
20 
30 
40 
50 


2919 
2929 
2940 
2951 
2961 
2972 


t 

4 

5 

1 


700 
705 
710 
715 
720 
724 


89 
66 
46 
28 
11 
97 


5202 
5217 
5232 
5248 
5261 
5276 


4 
3 

1 
9 
7 
5 


35° 

10 
20 
30 
40 
50 


1806 
1815 
1824 
1834 
1843 
1852 


6 

7 
9 

1 
3 
5 


278 
280 
283 
286 
289 
292 


05 
82 
60 
39 
20 
02 


3445 
3461 
3477 
3493 
3509 
3525 


9 
8 
7 
5 
4 
3 


2373-3 
2383-1 
2392.8 
2402.6 
2412.4 
2422-3 


472 
475 
479 
483 
487 
490 


08 
82 
59 
37 
16 
98 


4385 
4400 
4416 
4431 
4446 
4462 


3 

7 
1 
4 
8 
2 


55° 
10 
20 
30 
40 
50 


2982 
2993 
3003 
3014 
3025 
3035 


7 
3 
9 
5 
2 
8 


729 
734 
739 
744 
749 
754 


85 
76 
68 
62 
59 
57 


5291 

5306 

5320 

5335. 

5350. 

5365- 


3 

1 
9 
6 
4 
1 


36° 

> 10 
20 
30 
40 
50 


1861 
1870 
1880 
1889 
1898 
1907 


7 
9 

1 
4 
6 
9 


294 
297 
300 
303 
306 
309 


86 
72 
59 
47 
37 
29 


3541 
3557 
3572 
3588 
3604 
3620 


1 

8 
6 
5 
3 


46° 

10 
20 
30 
40 
50 


2432-1 
2441-9 
2451.8 
2461-7 
2471.5 
2481.4 


494 
498 
502 
506 
510 
514 


82 
67 
54 
42 
33 
25 


4477 
4492 
4508 
4523 
4538 
4554 


5 
8 

2 
5 
8 

1 


56° 

10 
20 
30 
40 
50 

57° 
10 
20 
30 
40 
50 


3046 
3057 
3067 
3078 
3089 
3100 


5 
2 
9 
7 
4 
2 


759 
764 
769 
774 
779 
784 


58 
61 
66 
73 
83 
94 


5379 

5394. 

5409. 

5423. 

5438. 

5453. 


8 
5 
2 
9 
6 
3 


37° 

10 
20 
30 
40 
50 


1917 
1926 
1935 
1945 
1954 
1963 


1 
4 
7 

3 
6 


312 
315 
318 
321 
324 
327 


22 
17 
13 
11 
11 
12 


3636 
3651 
3667 
3683 
3699 
3715 


1 
9 
7 
5 
3 



47° 
10 
20 
30 
40 
50 

48° 
10 
20 
30 
40 
50 

49° 

10 
20 
30 
40 
50 

50° 

10 
20 
30 
40 
50 
51° 


2491-3 
2501-2 
2511-2 
2521-1 
2531-1 
2541-0 


518 
522 
526 
530 
534 
538 


20 
16 
13 
13 
15 
18 


4569 
4584 
4599 
4615 
4630 
4645 


4 
7 
9 
2 
4 
7 


3110 
3121 
3132 
3143 
3154 
3165 


9 
7 
6 

4 
2 

1 


790 
795 
800 
805 
810 
816 


08 
24 
42 
62 
85 
10 


5467. 
5482- 
5497 
5511 
5526 
5541 


9 
5 
2 
8 
4 



38° 
10 
20 
30 
40 
50 


1972 
1982 
1991 
2000 
2010 
2019 


9 
2 
5 
G 
2 
6 


330 
333 
336 
339 

342 
345 


15 
19 
25 
32 
41 
52 


3730 
3746 
3762 
3778 
3793 
3809 


8 
5 
3 

8 
5 


2551-0 
2561-0 
2571-0 
2581-0 
2591-1 
2601.1 


542 
546 
550 
554 
558 
562 


23 
30 
39 
50 
63 
77 


4660 
4676 
4691 
4706 
4721 
4736 


9 
1 
3 
5 
7 
9 


58° 
10 
20 
30 
40 
50 


3176 
3186 
3197 
3208 
3219 
3230 



9 
8 
8 

7 
7 


821 
826 
831 
837 
842 
848 


37 
66 
98 
31 
67 
06 


5555 
5570 
5584 
5599 
5613 
5628 
5642 
5657 
5671 
5686 
5700 
5715 


6 
2 
7 
3 
8 
3 


39° 

10 
20 
30 
40 
50 


2029 
2038 
2047 
2057 
2066 
2076 



4 
8 
2 
6 



348 
351 
354 
358 
361 
364 


64 
78 
94 
11 
29 
50 


3825 
3840 
3856 
3872 
3888 
3903 


2 
9 
6 
3 

6 


2611.2 
2621.2 
2631.3 
2641-4 
2651-5 
2661-6 


566 
571 
575 
579 
583 
588 


94 
12 
32 
54 
78 
04 


4752 
4767 
4782 
4797 
4812 
4827 


1 
3 
4 
5 
7 
8 


59° 

10 
20 
30 
40 
50 


3241 
3252 
3263 
3274 
3285 
3296 


7 
7 
7 
8 
8 
9 


853 
858 
864 
869 
875 
880 


46 
89 
34 
82 
32 
84 


8 
3 
8 
3 
8 
2 


40° 

10 
20 
30 
40 
50 


2085 
2094 
2104 
2113 
2123 
2132 


4 
9 
3 
8 
3 
7 


367 
370 
374 
377 
380 
384 


72 
95 
20 
47 
76 
06 


3919 
3935 
3950 
3966 
3981 
3997 
4013 


3 

6 
3 
9 
5 
1 


2671-8 
2681-9 
2692-1 
2702-3 
2712-5 
2722-7 


592 
596 
600 
605 
609 
614 


32 
62 
93 
27 
62 
00 


4842 
4858 
4873 
4888 
4903 
4918 


9 

1 
2 
2 
-3 


60° 

10 
20 
30 
40 
50 
61° 


3308 
3319 
3330 
3341 
3352 
3363 



1 
3 
4 
-6 
-8 


886 
891 
897 
903 
908 
914 


38 
95 
54 
15 
79 
.45 


5729 
5744 
5758 
5-772 
5787 
5801 


7 

1 
5 
9 
3 

.7 


41° 


2142 


2 


387 


38 


2732-9 


618 


39 


4933 


-4 


3375 


.0 


920 


.14 


5816 






569 



TABLE II.— TANGENTS, EXTERNAL DISTANCES, AND LONG CHORDS 

FOR A 1° CURVE. 



A 


T^g. 


Ext. 
Dist. 
E, 


Long 
Chord 


A 


T^g= 


Ext. 

Dist. 


Lonff 
Chord 
iC. 


A 

81° 
10 
20 
30 
40 
50 

83° 
10 
20 
30 
40 
50 

83° 
10 
20 
30 
40 
50 

84° 
10 
20 
30 
40 
50 

85° 
10 
20 
30 
40 
50 


T^5' 


Ext. 
Dist. 


Long 
Chorcl 


61° 

10' 
20 
30 
40 
50 


3375.0 
3386.3 
3397.5 
3408.8 
3420.1 
3431.4 


920.14 
925.85 
931.58 
937.34 
943.12 
948-92 


5816-0 
5830-4 
5844-7 
5859-1 
5873-4 
5887-7 
5902-0 
5916-3 
5930.5 
5944.8 
5959-0 
5973-3 


71° 
10 
20 
30 
40 
50 

73° 
10 
20 
30 
40 
50 

73° 
10 
20 
30 
40 
50 

74° 
10 
20 
30 
40 
50 

75° 
10 
20 
30 
40 
50 

76° 
10 
20 
80 
40 
50 

77° 
10 
20 
30 
40 
50 

78° 
10 
20 
30 
40 
50 

79° 
10 
20 
30 
40 
50 

80° 
10 
20 
30 
40 
50 

81° 


4086-9 
4099-5 
4112-1 
4124-8 
4137-4 
4150-1 


1308-2 
1315-5 
1322-9 
1330-3 
1337-7 
1345-1 


6654-4 
6668-0 
6681-6 
6695.1 
6708.6 
6722-1 


4893-6 
4908-0 
4922-5 
4937-0 
4951-5 
4966-1 


1805-3 
1814-7 
1824-1 
1833.6 
1843.1 
1852-6 


7442-2 
7454-9 
7467-5 
7480-2 
7492-8 
7505-4 


03° 
10 
20 
30 
40 
50 


3442.7 
3454.1 
3465-4 
3476.8 
3488.2 
3499-7 
3511.1 
3522-6 
3534-1 
3545-6 
3557-2 
3568-7 


954-75 
960-60 
966-48 
972-39 
978-31 
984-27 


4162-8 
4175-6 
4188-4 
4201-2 
4214-0 
4226-8 


1352-6 
1360-1 
1367-6 
1375-2 
1382-8 
1390-4 


6735-6 
6749-1 
6762-5 
6776-0 
6789-4 
6802-8 

6816-3 
6829-6 
6843-0 
6856-4 
6869-7 
6883-1 


4980-7 
4995-4 
5010-0 
5024-8 
5039-5 
5054-3 


1862-2 
1871-8 
1881-5 
1891-2 
1900-9 
1910-7 


7518-0 
7530-5 
7543-1 
7555-6 
7568-2 
7580-7 


63° 

10 
20 
30 
40 
50 


990.24 
996-24 
1002-3 
1008-3 
1014.4 
1020-5 


5987-5 
6001.7 
6015.9 
6030.0 
6044.2 
6058.4 


4239-7 
4252-6 
4265-6 
4278-5 
4291-5 
4304-6 


1398-0 
1405-7 
1413-5 
1421-2 
1429-0 
1436-8 


5069-2 
5084-0 
5099-0 
5113-9 
5128-9 
5143-9 


1920-5 
1930-4 
1940-3 
1950-3 
1960-2 
1970-3 
1980-4 
1990-5 
2000-6 
2010-8 
2021-1 
2031-4 


7593-2 
7605.6 
7618.1 
7630.5 
7643-0 
7655-4 


64° 

10 
20 
30 
40 
. 50 


3580-3 
3591-9 
3603-5 
3615-1 
3626-8 
3638-5 


1026-6 
1032-8 
1039.0 
1045.2 
1051.4 
1057-7 


6072.5 
6086.6 
6100.7 
6114.8 
6128.9 
6143-0 


4317-6 
4330-7 
4343 - 8 
4356-9 
4370-1 
4383-3 
4396-5 
4409 . 8 
4423 - 1 
4436-4 
4449 - 7 
4463 - 1 
4476-5 
4489-9 
4503-4 
4516-9 
4530-4 
4544-0 


1444-6 
1452-5 
1460-4 
1468-4 
1476-4 
1484-4 


6896-4 
6909-7 
6923-0 
6936-2 
6949 - 5 
6962-8 


5159-0 
5174-1 
5189-3 
5204-4 
5219-7 
5234-9 


7667-8 
7680-1 
7692.5 
7704-9 
7717-2 
7729-5 


65° 

10 
20 
30 
40 
50 


3650-2 
3661.9 
3673.7 
3685.4 
3697.2 
3709-0 


1063-9 
1070-2 
1076-6 
1082-9 
1089-3 
1095-7 
1102-2 
1108-6 
1115.1 
1121.7 
1128-2 
1134-8 


6157.1 
6171.1 
6185.2 
6199-2 
6213-2 
6227-2 


1492-4 
1500-5 
1508-6 
1516-7 
1524-9 
1533-1 


6976-0 
6989-2 
7002-4 
7015-6 
7028-8 
7041-9 


5250-3 
5265-6 
5281-0 
5296-4 
5311-9 
5327-4 


2041-7 
2052-1 
2062-5 
2073-0 
2083-5 
2094-1 


7741.8 
7754.1 
7766.3 
7778-e 
7790. & 
7803-0 


66° 

10 
20 
30 
40 
50 


3720-9 
3732-7 
3744-6 
3756.5 
3768.5 
3780.4 
3792.4 
3804.4 
3816.4 
3828.4 
3840.5 
3852.6 
3864.7 
3876.8 
3889.0 
3901.2 
3913.4 
3925-6 


6241-2 
6255-2 
6269-1 
6283-1 
6297-0 
6310-9 


1541-4 
1549-7 
1558-0 
1566-3 
1574-7 
1583-1 


70550 
7068-2 
7081-3 
7094-4 
7107-5 
7120-5 

7133-6 
7146-6 
7159-6 
7172-6 
7185-6 
7198-6 
7211-6 
7224-5 
7237-4 
7250-4 
7263-3 
7276-1 


86° 
10 
20 
30 
40 
50 

87° 
10 
20 
30 
40 
50 

88° 
10 
20 
30 
40 
50 


5343-0 
5358-6 
5374-2 
5389-9 
5405-6 
5421-4 


2104-7 
2115-3 
2126-0 
2136-7 
2147-5 
2158-4 


7815.2 
7827-4 
7839.6 
7851-7 
7863-8 
7876-0 


67° 

10 
20 
30 
40 
50 


1141-4 
1148-0 
1154-7 
1161-3 
1168 1 
1174-8 


6324-8 
6338-7 
6352-6 
6366-4 
6380-3 
6394-1 


4557-6 
4571-2 
4584-8 
4598-5 
4612-2 
4626-0 


1591-6 
1600-1 
1608-6 
1617-1 
1625-7 
1634-4 
1643-0 
1651-7 
1660-5 
1669-2 
1678-1 
1686-9 


5437-2 
5453-1 
5469-0 
5484-9 
5500-9 
5517-0 


2169-2 
2180-2 
2191-1 
2202-2 
2213-2 
2224-3 


7888.1 
7900-1 
7912-2 
7924-3 
7936-3 
7948-3 


68° 
10 
20 
30 
40 
50 


1181-6 
1188-4 
1195-2 
1202-0 
1208-9 
1215-8 
1222-7 
1229-7 
1236-7 
1243-7 
1250-8 
1257-9 


6408-0 
6421-8 
6435-6 
6449-4 
6463-1 
6476-9 


4639-8 
4653-6 
4667-4 
4681-3 
4695-2 
4709-2 


55331 
5549-2 
5565-4 
5581-6 
5597-8 
5614-2 


2235-5 
2246-7 
2258-0 
2269-3 
2280-6 
2292.C 


7960-3 
7972-3 
7984-2 
7996-2 
8008-1 
80200 


69° 

10 
20 
30 
40 
50 


3937-9 
3950.2 
3962.5 
3974.8 
3987.2 
3999-5 


6490-6 
6504-4 
6518-1 
6531-8 
6545-5 
6559-1 


4723-2 
4737-2 
4751-2 
4765-3 
4779-4 
4793-6 


1695-8 
1704-7 
1713-7 
1722-7 
1731-7 
1740.8 


7289-0 
7301-9 
7314-7 
7327-5 
7340-3 
7353-1 
7365-9 
7378-7 
7391-4 
7404-1 
7416-8 
7429-5 


89° 
10 
20 
30 
40 
50 

90° 
10 
20 
30 
40 
50 


5630-5 
5646-9 
5663-4 
5679-9 
5696-4 
57130 
5729-7 
5746-3 
5763-1 
5779-9 
5796-7 
5813-6 


2303-5 
2315-0 
2326-6 
2338-2 
2349-8 
2361-5 


8031-9 
8043 -8 
8055-7 
8067-5 
8079-3 
8091-2 


70° 

10 
20 
30 
40 
50 


4011-9 
4024.4 
4036-8 
4049 . 3 
4061.8 
4074-4 


1265-0 
1272.1 
1279.3 
1286-5 
1293-7 
1300-9 


6572-8 
6586-4 
6600.1 
6613.7 
6627.3 
6640-9 


4808-7 
4822-0 
4836-2 
4850-5 
4864-8 
4879-2 


1749-9 
1759-0 
1768-2 
1777-4 
1786-7 
1796-0 


2373-3 
2385-1 
2397-0 
2408-9 
2420-9 
2432-9 


8103-0 
8114-7 
8126-5 
8138-2 
8150-0 
8161.7 


711 


4086.9 


1308-2 


6654-4 


4893.6 1805-3 


7442 - 2 


91° 


5830-5 


2444-9 


8173.4 



57u 



TABLE III.— SWITCH-LEADS AND DISTANCES. 
LEAD-RAILS CIRCULAR THROUGHOUT; GAUGE 4' 8l". Sec §262. 

















Length of 




Frog 

No, 


Frog Angle 


Lead (L) 
(Eq. 79) 


Chord 
iQT) 


Radius of 
Lead-rails 


Log r. 


Switch- 
rails 


Frog 
No. 


in) 


(F). 




(Eq. 77). 


(r,Eq.78). 




{QK, 


in) 
















Eq. 81). 




4 


14° 15' 


00" 


37.67 


37.38 


150.67 


2.1780T 


11.73 


4 


5 


11 25 


16 


47.08 


46.85 


235.42 


.37183 


14.65 


5 


6 


9 31 


38 


56-50 


56.30 


339.00 


.53020 


17.62 


6 


7 


8 10 


16 


65.92 


65.75 


461.42 


.66409 


20.53 


7 


8 


7° 09' 


10 


75.33 


75.19 


602.67 


.78007 


23.48 


8 


9 


6 21 


35 


84.75 


84.62 


762.75 


.88238 


26.43 


9 


9i 


6 01 


32 


89.46 


89.33 


849.85 


.92934 


27.97 


9i 


10 


5 43 


29 


94.17 


94.05 


941.67 


2.97389 


29.37 


10 


11 


5° 12' 


18" 


103.58 


103.47 


1139.42 


3.0566O 


32.31 


11 


12 


4 46 


19 


113.00 


112.90 


1356.00 


. 13-526 


35.25 


12 


15 


3 49 


06 


141.25 


141.17 


2118.75 


.32608 


44.06 


15 


16 


3 34 


47 


150.67 


150.59 


2410.67 


.38213 


46-98 


16 


18 


3° 10' 


56" 


169.50 


169.43 


3051.00 


.48444 


52.87 


18 


20 


2 51 


51 


188.33 


188.27 


3766.67 


.57595 


58.76 


20 


24 


2° 23' 


13" 


226.00 


225.95 


5424.00 


3.73432 


70.52 


24 



TURNOUTS WITH STRAIGHT POINT-RAILS AND STRAIGHT 
FROG RAILS; GAUGE 4' 8i". See §§ 265 and 276- 





1 


Frog. 


Switch. 


Switch Dimensions. 


6 


bed 
.S3 
^^ 

CO 


(HK) 


a 
(DM) 


Angle. 
(«) 


Radius, 
(r) 


Degree 

of 
Lead 
Curve. 

(D) 


L'gth. 


Closure. 


in) 


Str'ght 
Rail. 


Curv'd 
Rail. 


4 
5 
6 
7 


ft. 

0.17 
0.21 
0-25 
0.29 


ft. in. 
3 2 

3 7 

4 
4 5 


ft. in. 
8 6 

10 

11 

12 6 


ft. in. 
11 
11 
11 
16 6 


/ // 

2 36 19 
2 36 19 
2 36 19 
1 44 11 


ft. 
112.26 
183.22 
273.95 
364.88 


/ // 

52 53 56 
31 40 24 
21 01 58 
15 47 19 


ft. 

37.22 
42.98 
48.36 
62.23 


ft. 
22.88 
28.19 
33.11 
41.02 


ft. 
23.29 
28-55 
33-38 
41.24 


8 
9 

91 
10 


0.33 
0.37 
0.40 
0.42 


4 9 
6 
6 
6 


13 6 
16 
16 
16 6 


16 6 
16 6 
16 6 
16 6 


1 44 11 
1 44 11 
1 44 11 
1 44 11 


488.71 
616.27 
699.97 
790.25 


11 44 40 
9 18 27 
8 11 33 
7 15 18 


67.80 
72.61 
75-30 
77.93 


46.22 
49.74 
52.40 
55.01 


46.42 
49.92 
52.58 
55.17 


11 
12 
15 
16 


0.46 
0.50 
0.62 
0.67 


6 

6 5 

7 8 

8 


17 6 

18 6 
22 6 
24 


22 
22 
33 
33 


1 18 08 
1 18 08 
52 05 
52 05 


940.21 
1136-34 
1744-38 
2005.98 


6 05 48 
5 02 38 
3 17 01 
2 51 24 


92.52 

97.75 

133.64 

136.62 


64.06 
68.83 
92.36 
94.95 


64.20 
68.96 
92.46 
95.05 


18 
20 
24 


0.75 
0.83 
1.00 


8 10 

9 8 
11 4 


26 6 
29 
34 6 


33 
33 
33 


52 05 
52 05 
52 05 


2587.66 
3262.98 
4932.77 


2 12 52 
1 45 22 
1 09 42 


147.13 
157.18 
176.09 


104.54 
113.68 
130.66 


104.61 
113.76 
130.77 



571 



TABLE III. SWITCH LEAD AND DISTANCES.— Con^mz^ed. 
TRIGONOMETRICAL FUNCTIONS OF THE FROG ANGLES. 



Frog 
No. 
(n) 


Frog Angle 


Nat. 
sinF. 


Nat. 
cos F. 


Log 

sini^. 


Log 
cos F. 


Log 

cot i^. 


Log 
vers F. 


Frog 
No. 

(n) 


4 
5 
6 
7 


14° 15' 00'' 

11 25 16 

9 31 38 

8 10 16 


.24615 
.19802 
.16552 
.14213 


.96923 
.98020 
.98621 
.98985 


9.39120 
.29670 
.21884 
.15268 


9.98642 
.99131 
.99397 
.99557 


10.59522 
.69461 
.77513 
.84288 


8.4881T 
.29670 
.13966 

8.00655 


4 
5 
6 
7 


8 

9 

10 


7 09 10 
6 21 35 
6 01 32 
5 43 29 


.12452 
.11077 
.10497 
.09975 


.99222 
.99385 
.99448 
.99501 


.09522 

. 04442 

9.02107 

8.99891 


.99660 
.99732 
.99759 
.99783 


. 90138 

.95289 

.97652 

10.99892 


7.89110 
.78915 
.74232 
.69787 


8 

U 

10 


11 
12 
15 
16 


5 12 18 
4 46 19 
3 49 06 
3 34 47 


.09072 
.08319 
.06659 
.06244 


.99588 
.99653 
.99778 
.99805 


.95770 
.92007 
.82343 
.79543 


.99820 
.99849 
.99903 
.99915 


11.04050 
.07842 
.17560 
.20370 


.61527 
.53986 
.34631 
.29028 


11 

12 
15 
16 


18 
20 
24 


3 10 56 
2 51 51 
2° 23' 13" 


.05551 
.04997 
.04165 


.998^6 
.99875 
.99913 


.74438 

.69869 

8.61959 


.99933 

.99945 

9.99962 


.25494 

.30076 

11.38003 


.18807 
7.09663 
6.93834 


18 

20 
24 



672 



TABLE IV —ELEMENTS OF TRANSITION CURVES. 



o 


fHcqco^lr5<!D^*ooo50 


^ 


ooo5ir>io«ooo<Nco 

OO050500C0C0t<.t^CN 

ooa3a>a>03CJ>ooc^co 


I— 1 1— 1 rH i-H CN CN 


55 

fao 

o 
-J 


lOOIlOli— liOICNIC^ I— 1 '^ ■^li— 1 
COCDOOOOOCDt^rHCDCD 
lOr^rHCNCO^i— llOr-lO 
COCOCOr-lt>.0500rt<05CN 
rj<i— (lOOSi— icoir>i>-ooo 


OOOSOSOSOOOOOrH 


8 


C«-CDCN00O.-lt<-r-ICq03 
CMCOOO.-IOOO.-ICOCDCO 

Oi— icooom'<:l<ooir5i>-^ 


oooorHcqcoinc-o 

1-H 


-e- 


irjiloo ^ CO OS I— iiCMicq c^ in 
inc^ooiocoosc^ojcDt^ 

CDOCNCDOOOOOOCNCO 

t>.eocoo.csic<it-oooom 

C0C005C0I>-OCSl^CD00 


^mmcocot>t^t>c>-t^ 


i 


O 05ICDia>ICDI^ 05 CDIOICO 

ocjsoiooc^iOrHCDcnoo 

OCT>050305030500t^CD 
O 03 05 05 05 Oi 03 03 03 03 
O 03 Oi C3 03 03 03 OJ 03 03 


0a)030>03a>03030303 
1-H 


-e- 

fciD 


lOOIOKMliO Olinit-I'^ OIO 
t>.O3O3C>.0000CDCDCOrH 

eoi— ii— leoI-^cD00O3O^t^- 
eo00I— icoiocotxoooso 


C>*C^000000C»C00000O3 




03lt«»l"«*l03lrH 03 (M 00 
0) C 0303030000 CD in CM 
C (^0303030303030303 
O C 0303030303030303 


z 


cqiin I-H 00 1". ooioi-^t* o t- 

CqcDCOi— iCMlDi— I0O0OO3 
OOi-ICqC0^CDt^03.-l 
OOOOOOOOOi-l 




Total 
Central Angle 


oooooooooo 
coeooococoooeooo 

"[><Mmir>cqc-oot>-c<j 
oc<i'^r-Hmcocoeocoio 

OOOrHi-HCNeO'«*lOCD 




r-* 



o 


o o eg ID 

.-1 CO CM -«* 

00 <N -«# CO 
CM rH lO CO 


CO 

o 


§ 

in 


lO 

CD 


in 

rH 
CO 


o 

00 


o o 

CO o 

t- o 
CO o 




rJ<'5i<COCOCO(NICgrH.-lO 


O 




"locNOt^mcvJOt-in 

•«i<C<ICOOrHlf30COo* 


o 
o 


o 

CO 


05 


oo-^t-oocDi-iininco 

COCVJO^CJOCOOCO 


o 

o 


co 




COCOCOCSICSlCgrHi-HO 


o 


o 




cqiot-ocMiot-o 

(Ml— ICOCOID'^OO 


o 

o 


in cq 

Til eg 


00 


%iii-iir3is.cDcoooo 
lOTjicMO-^cginco 


o 

o 


CO 03 
CO o 




(MCq<M<Nr-Hi-HOO 


o 


O rH 




o C"- in <N o c»- in 

O O "5*4 in CO CO rH 


o 
o 


o eg in 

O in rH 


r^ 


in CO 00 I-H eg o CO 

r-H O "^ CO 1-1 in CSJ 


o 
o 


O iH CO 
CO O CO 




(N eg rH rH I-H O O 


o 


O rH rH 




V- c eg in t- o 
CO o in I-H o eo 


o 
o 


in eg o t* 
rH eg o o 


o 


o o CD rH CO eg 

"«;14 CO rH O •«* Cq 


o 
o 


CO ^ in 00 
eg in cq in 




rH rH pH I-H O O 


o 


O O rH rH 




"in eg o C"- in 

rH in O CO ■'il 


o 
o 


o eg in c<- o 
CO in Ti< o o 


10 


Vi I-H o in 00 

rH O m CO rH 


§ 


eg CO CO CO in 

eg ■«*» rH Td< rH 




o 

rH rH O O O 


o 


o o rH rH eg 




eg in t^ o 

in ^ o o 


o 
o 


in eg o t^ in eg 
■^ eg CO o rH in 


'^ 


"cD 00 00 in 

'ii^ CO eg r-i 


o 
o 


00 03 eg 00 CD CO 
rH CO o eg in eg 




o o o c 


o 


o 


o 


ft 


t-i 


rH 


e. 



o 

CO 


CO 


m 


§ 


eg 


o 
eg 




o 
o 


o 


O 


c 


o 



o o o rH rH eg eg 



o 


c 

CO 


o 

c 


CO 


o 


c 

c 


o 

o 


o 


o 



oooorHrHcgeg 



o 



eg 
in 


in 


o 


o 
o 


eg 
eg 


in 

rH 


00 


o 

00 


CD 


oo 
eg 


CO 


o 
o 


03 


f* 

^ 


in 

o 


eg 
oo 



o o o o rH rH rH eg eg 



in 


eg 
eg 


o 

CO 


o 


m 


eg 
m 


o 
o 


CO 


in o 

■«^ eg 


CO 


03 


t^ 


00 

eg 




CD 

in 


in 


in 

CO 


00 ^ 
m eg 


o 


o 


o 


o 


o 


o 


rH 


i-\ 


>-i eg 



C^r^WW^I0O^•Q005© 



573 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 
Dimensions of various 0° 30'-per-25-feet spirals. — Part B. 



Deg. 
of 


L^^th 


ZK 


QK 


A'N 


NQ 


A' to Z, 


""^r- 








curve. 


spiral 










Defl. 


Dist. 


curve. 


1° 00' 


25 


0.03 


25.00 


0.01 


12.50 


0° 


04' 


12.50 


P 00' 


10 


50 


0.14 


50.00 


0.03 


17.86 





11 


32.14 


10 


20 


50 


0.14 


50.00 


0.04 


21.88 





11 


28.13 


20 


30 


50 


0.14 


50.00 


0.05 


25.00 





11 


25.00 


30 


40 


75 


0.38 


75.00 


0.09 


30.00 





22i 


45.00 


40 


50 


75 


0.38 


75.00 


0.11 


43.09 





22^ 


40.91 


50 


S'' 00' 


75 


0.38 


75.00 


0.14 


37.50 


0° 


22y 


37.50 


2° 00' 


10 


100 


0.82 


100.00 


0.19 


42.30 





37i 


57.69 


10 


20 


100 


0.82 


100.00 


0.23 


46.43 





37^ 


53.57 


20 


30 


100 


0.82 


100.00 


0.27 


50.00 





37i 


50.00 


SO 


40 


125 


1.50 


124.99 


0.35 


54.68 





56 


7C.31 


40 


50 


125 


1.50 


124.99 


0.42 


58.81 





56 


66.18 


50 


8* 00' 


125 


1.50 


124.99 


0.48 


62.49 


0° 


56' 


62.50 


3° 00' 


10 


150 


2.48 


149.97 


0.58 


67.09 




19 


82.59 


10 


20 


150 


2.48 


149.97 


0.68 


71.24 




19 


78.75 


20 


30 


150 


2.48 


149.97 


0.76 


74.98 




19 


75.00 


30 


40 


175 


3.82 


174.93 


0.90 


79.52 




45 


95.45 


40 


50 


175 


3.82 


174.93 


1.03 


83.67 




45 


91.30 


50 


4« 00' 


175 


3.82 


174.93 


1.15 


87.47 


l** 


45' 


87.50 


4'' 00' 


10 


200 


5.56 


199.87 


1.32 


91.96 


2 


15 


108.00 


10 


20 


200 


5.56 


199.87 


1.48 


96.11 


2 


15 


103.85 


20 


SO 


200 


5.56 


199.87 


1.63 


99.95 


2 


15 


100.00 


30 


40 


225 


7.79 


224.77 


1.88 


104.40 


2 


49 


120.54 


40 


50 


225 


7.79 


224.77 


2.08 


108.55 


2 


49 


116.38 


50 


6° 00' 


225 


7.79 


224.77 


2.27 


112.42 


2° 


49' 


112.50 


5° 00' 


10 


250 


10.49 


249.62 


2.51 


116.83 


3 


26 


133.06 


10 


20 


250 


10.49 


249.62 


2.76 


120.98 


3 


26 


128.91 


20 


SO 


250 


10.49 


249.62 


3.00 


124.88 


3 


26 


125.00 


80 



574 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 
Dimensions of various 0° 30'-per-25-feet spirals. — Part C. 

Values of AN, 



A' 






Degree of curve. 






1° 


2° 


3° 


4° 


5° 


5° SO' 


20 


0.00 


0.00 


0.01 


0.02 


0.04 


0.05 


4 


0.00 


0.01 


0.02 


0.04 


0.08 


0.10 


6 


0.00 


0.01 


0.03 


0.06 


0.12 


0.16 


8 


0.00 


0.01 


0.03 


0.08 


0.16 


0.21 


10 


0.00 


0.01 


0.04 


0.10 


0.20 


0.26 


12° 


0.00 


0.01 


0.05 


0.12 


0.24 


0.32 


14 


0.00 


0.02 


0.06 


0.14 


0.28 


0.37 


16 


0.00 


0.02 


0.07 


0.16 


0.32 


0.42 


18 


0.00 


0.02 


0.08 


0.18 


0.36 


0.47 


20 


0.00 


0.02 


0.08 


0.20 


0.40 


0.53 


22° 


0.00 


0.03 


0.09 


0.22 


0.44 


0.58 


24 


0.00 


0.03 


0.10 


0.24 


0.48 


0.64 


26 


0.00 


0.03 


0.11 


0.26 


0.52 


0.69 


28 


0.00 


0.03 


0.12 


0.29 


0.57 


0.75 


SO 


0.00 


0.04 


0.13 


0.31 


0.61 


0.80 


32° 


0.00 


0.04 


0.14 


0.33 


0.65 


0.86 


84 


0.00 


0.04 


0.15 


0.35 


0.69 


0.92 


86 


0.00 


0.04 


0.16 


0.37 


0.74 


0.97 


88 


0.00 


0.05 


0.16 


0.39 


0.78 


1.03 


40 


0.00 


0.05 


0.17 


0.42 


0.83 


1.09 


42° 


0.00 


0.05 


0.18 


0.44 


0.87 


1.13 


44 


0.00 


0.06 


0.19 


0.46 


0.92 


1.21 


46 


0.00 


0.06 


0.20 


0.49 


0.96 


1.27 


48 


0.00 


0.06 


0.21 


0.51 


1.01 


1.34 


60 


0.00 


0.06 


-0.22 


0.53 


1.06 


1.40 


62° 


0.00 


0.07 


0.23 


0.56 


1.11 


1.48 


64 


0.01 


0.07 


0.24 


0.58 


1.16 


1.53 


66 


O.OI 


0.07 


0.25 


0.61 


1.21 


1.59 


68 


0.01 


0.08 


0.26 


0.64 


1.26 


1.68 


60 


0.01 


0.08 


0.28 


0.66 


1.31 


1.73 


62° 


0.01 


0.08 


0.29 


0.69 


1.36 


1.80 


64 


0.01 


0.09 


0.30 


0.72 


1.42 


1.87 


66 


0.01 


0.09 


0.31 


0.74 


1.47 


1.95 


68 


0.01 


0.09 


0.32 


0.77 


1.53 


2.02 


TO 


0.01 


0.10 


0.33 


0.80 


1.59 


2.10 


T2° 


0.01 


0.10 


0.35 


0.83 


1.65 


2.18 


74 


0.01 


0.10 


0.36 


0.83 


1.71 


2.26 


76 


0.01 


0.11 


0.37 


0.90 


1.77 


2.34 


78 


0.01 


0.11 


0.39 


0.93 


1.84 


2.43 


80 


0.01 


0.11 


0.40 


0.96 


1.91 


2.51 


82«» 


0.01 


0.12 


0.42 


1.00 


1.97 


2.60 


84 


0.01 


0.12 


0.43 


1.03 


2.04 


2.70 


86 


0.01 


0.13 


0.45 


1.07 


2.12 


2.79 


88 


0.01 


0.13 


0.46 


1.10 


2.19 


2.89 


eo 


0.01 


0.14 


0.48 


1.15 


2.27 


3.00 


92° 


0.01 


0.14 


0.49 


1.19 


2.35 


3.10 


94 


0.01 


0.15 


0.51 


1.23 


2.44 


3.21 


96 


0.01 


0.15 


0.53 


1.27 


2.52 


3.33 


98 


0.01 


0.16 


0.55 


1.32 


2.61 


3.45 


100 


0.01 


0.16 


0.57 


1.37 


2.71 


3.57 



575 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 



'o 




$Si 


OCD^aXMlOCMOOt^ 


rH 1— 1 1— 1 1— 1 cq cq 


8 

o 

—J 


ICMICOICSJIOOlt^ 00 r-iia> 05II-1 

COCOCNCOCOLOCNIOOO 
COCOOOi— It^OiOO^OXN 


000503OOOOrHrHrH 


8 


incoeocoojooob^cqco 

IOI>.CDC003CDCSJOO.-I 

ocqi>-coa)05CDr-iioa> 


rH Cq -^ t* rH ID O 
fHrHCM 


> 

bo 
o 

—1 


lO ««:1< OOICO CO -«* OOli-HICOl"^ 

cDcoooiocot>-coeocDcq 
oocgrjHooocMCNOeot-. 
i>-eococ-coc<jt^0300io 

0503ir)0500COOOOCN'>!d< 


^lOCOCOt«*I>I>0000C0 


—I 


I05ICD lOlOO t>-IO ID c>qit> t^ 

oiosoomorHt^coineo 

05 05 05 05 CO 00 CD <^i< i-H t- 
03 O) 03 Oi 03 O) 03 03 03 00 

05CT>a>05a)a>a>05CT>03 


OD CO O) 03 OS O) OS O) OS 03 


bo 


IrHICq Cq 00 O C0I03 COICOIO 
0005C^CDCD•<;t^00C0CNIO 
05COt.05U:>.-HO<^OCO 
eOrHi-ieOi-HCO00O3O5C^ 
CDi-H^CD00050i— ICqCO 


!>■ 00 00 00 00 00 03 03 03 05 . 


-e- 

o 

z 


ositoioioo ooiio t- OOICO 
Oo3a)a>i>.iocsit>.orH 

rtO3O5O5O3O3a)0000t^ 
O 03 03 Od 03 03 03 Oi 05 O) 




15 


ICO i-H <M CO ^ lOI00lrJ< r-i !>. 
Tj<COCDCOlOi-li— ICOlOt^ 

Oi— icqrjicooicqioosco 

OOOOOOrHr-lrHCSI 




Total 
Central Angle 


"iomooioioooinio 

i-l'>*COeO"<i<rHOOr-H'^ 

o 

oorHcqeotot^osi— ICO 

1-11-4 


Point. 


f-H Cq CO td« U) CO D* 00 O) o 







V 
























cq 


lo 05 cq t> o lo o 


in 


O 


o 






C<I 


O 'il* CO rH O ^ CO 


r-i 


o 


O 




o 














1-1 


"co 


lo 00 b". rH o CO cq 


CO 


in 


o 






lO 


cq <* o cq 00 00 CO 


cq 




o 






_k 


00 C>. t"* CD lO Til CO 


*q 


t-i 


o 






"V^ 


t* cq in o in o lo 


o 


o 


o 






CO 


-«* O rH CO rf O .-1 


CO 


o 


o 




05 


"c^ 


00 lO CD cq CO O rH 


t* 


o 


in 






I— 1 


** rH CO lO O 1-1 i-l 


o 


o 


i-i 


1 




o 


CD CD IT) «* r}< CO cq 


rH 


o 


r-i 


i 




^ 


1 








c3 




!>■ 


o in o lo o in o 


O 


o 


in 






-^ 


CO .-1 O >«* CO rH O 


O 


CO 


-^ 




GO 


00 


cq rH in CO t-1 CD o 
cq to »H CO ^ in o 


o 


t* 


00 


i 




""^ 


o 


o 


i-{ 




o 
ID 


ID *# Til CO cq rH 1-1 


o 


r-i 


cq 


^) 




V 












M 




o 


in O ID O ID o 


o 


o 


in 


o 


w 




o 


•-< CO Til o 1-1 CO 


o 


o 


-* 


CO 


.s 


r^ 


o 


CO t> CO in 1-1 cq 


o 


o 


00 


cq 


o 




CO 


o CO o cq ^ in 


o 


o 


o 


rH 


,13 
















+3 




o 












i=! 




-<* 


•«* CO CO cq rH oioi 


r-t 


cq 


eo 


















o 


















^ 

^ 

Ti 




irj 


o in o in o 


o 


o 


in 


o 


in 




rH 


O Til CO rH o 


o 


CO 


-<# 


o 


>-i 


CO 


%H 


o CO cq CD in 


o 


cq 


00 


o 


CD 


s 




cq 


o CO o cq Til 


o 


in 


Til 


in 


in 


*s 




o 














3 




CO 


CO cq cq rH o 


o 


o 


_rH_ 


cq 


CO 


o 




V. 














8 




o 


in o in o 


o 


o in 


o 


in 


o 




CO 


Til O rH CO 


o 


O Til 


CO 


rH 


o 


-M 


«5 
















g 




cq 


CO O rH t^ 


o 


in CO 


!>. 


CD 


o 


'o 




cq 


O Til rH CO 


o 


** CO 


cq 


cq 


CO 


a 




o 














© 




cq 


CSi r-i r-i O 


o 


O rH 


cq 


oo 


•«;»< 


r^:: 




^ 














-M 




"lO 


o in o 


o 


o in o 


in 


o 


CO 


-tJ 




''il 


CO rH O 


o 


CO tH O 


rH 


CO 


-<* 


c3 


■«j1 
















•+3 




"co 


C^ CO o 


o 


IN 00 in 


CD 


cq 


CO 


1 




CO 


rH in 00 


o 


00 rH o 


in 


in 


in 




o 

r-J 


rH O O 


o 


o rH cq 


cq 


CO 


Til 




^ 












-tJ 




o 


in o o 


o in o in 


o 


in 


00 


<v 






rH CO O 


O Til CO rH 


o 


-^ 


cq 


si 


w 


U3 


rH cq o 


o CO cq CD 


m 


00 


t^ 


B 




lO 


tH cq o 


CO o -^ cq 


rH 


o 


o 


o 




o 


o o o 


O rH rH cq 


CO 


■^ 


in 




^ 












en 




"lO 


o 


o o in o in o 


in 


00 


rH 


fl 




r-i 


o 


O CO Til O rH CO 


Til 


m 


rH 


.2 


N 


*co 


in 


o cq 00 o CO b- 


CO 


Til 


r-i 


1 




cq 


1— 1 


o cq Til cq in CO 


cq 


l-l 


rH 


03 
CI 






o 


o 


O O O rH rH cq 


CO 


Tjl 


in 


p 




^ 
















o 


o 


o in o in o in 


o 


CO 


in 






CO 


o 


O rJl CO f-1 O Til 


CO 


r-i 


in 




H 


t^ 


o 


in CO c^ CO o 00 


cq 


f-i 


Til 






o 


o 


rH CO in cq o CO 


cq 


r-i 


O 






% 


o 


o o o rH cq cq 


CO 


Til 


in 






o 


o in o in o in o 


CO 


CD 


00 








CO •«* O rH CO Til o 


t-^ 


cq 


CO 




©t 


o 


t>- 00 in CO cq CO o 


r-{ 


tx. 


00 








o rH CO in cq in CO 


i-i 


in 


•^ 






o 
o 


O O O O rH rH cq 


CO 


CO 


Tjl 


bD 












^c 
























1- 


Oi 


H N W tH »0 O £>• 


QO 


05 


o 

iH 
















i^ 























576 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 
Dimensions of various l°-per-25-feet spirals. — Part B. 



Deg. 
of 


L'gth 


ZK 


QK 


A'N 


NQ 


A' to Z. 


^oT 


of 








curve. 


spiral 










Defl. 


Dist. 


curve. 


2° 00' 


25 


0.06 


25.00 


0.03 


12.50 


0° 


07^' 


12.50 


2° 00' 


10 


50 


0.27 


50.00 


0.05 


15.38 





22^ 


34.62 


10 


20 


50 


0.27 


50.00 


0.06 


17.86 





22^ 


32.14 


20 


30 


50 


0.27 


50.00 


0.08 


20.00 





22i 


30.00 


30 


40 


50 


0.27 


50.00 


0.09 


21.87 





22^ 


28.13 


40 


50 


50 


0.27 


50.00 


0.10 


23.53 





22^ 


26.17 


50 


8*» GO' 


50 


0.27 


50.00 


0.11 


25.00 


0° 


22V 


25.00 


3° GO' 


10 


75 


0.76 


74.99 


0.14 


27.63 





45 


47.37 


10 


20 


75 


0.76 


74.99 


0.17 


29.99 





45 


45.00 


20 


80 


75 


0.76 


74.99 


0.20 


32.13 





45 


42.86 


30 


40 


75 


0.76 


74.99 


0.23 


34.08 





45 


40.19 


40 


50 


75 


0.76 


74.99 


0.25 


35.86 





45 


39.13 


50 


4« GO' 


75 


0.76 


74.99 


0.27 


37.49 


0° 


45' 


37.50 


4° GO' 


10 


100 


1.64 


99.98 


0.33 


39.98 




15 


60.00 


10 


20 


100 


1.64 


99.98 


0.38 


42.29 




15 


57.69 


20 


30 


100 


1.64 


99.98 


0.42 


44.43 




15 


55.56 


30 


40 


100 


1.64 


99.98 


0.47 


46.41 




15 


53.57 


40 


50 


100 


1.64 


99.98 


0.51 


48.26 




15 


51.72 


50 


6« GO' 


100 


1.64 


99.98 


0.55 


49.98 


JO 


15' 


50.00 


5« GO' 


10 


125 


3.00 


124.94 


0.62 


52.39 




52^ 


72.58 


10 


20 


125 


3.00 


124.94 


0.70 


54.65 




52i 


70.31 


20 


30 


125 


3.00 


124.94 


0.77 


56.78 




52i- 


68.18 


30 


40 


125 


3.00 


124.94 


0.83 


58.79 




52^ 


66.18 


40 


50 


125 


3.00 


124.94 


0.90 


60.67 




52^ 


64.29 


50 


e« GO' 


125 


3.00 


124.94 


0.95 


62.46 


r 


52V 


62.50 


6° GO' 


10 


150 


4.96 


149.87 


1.06 


64.81 


2 


37i 


85.14 


10 


20 


150 


4.96 


149.87 


1.16 


67.04 


2 


37i 


82.89 


20 


30 


150 


4.96 


149.87 


1.26 


69.17 


2 


37i 


80.77 


30 


40 


150 


4.96 


149.87 


1.35 


71.18 


2 


37i 


78.75 


40 


50 


150 


4.96 


149.87 


1.44 


73.10 


2 


37i 


76.83 


50 


7° 00' 


150 


4.96 


149.87 


1.52 


74.92 


2° 


374' 


75.00 


7° GO' 


10 


175 


7.63 


174.72 


1.67 


77.23 


3 


30 


97.67 


10 


20 


175 


7.63 


174.72 


1.80 


79.44 


3 


30 


95.45 


20 


30 


175 


7.63 


174.72 


1.93 


81.55 


3 


30 


93.33 


30 


40 


175 


7.63 


174.72 


2.05 


83.58 


3 


30 


91.30 


40 


50 


175 


7.63 


174.72 


2.17 


85.51 


3 


30 


89.36 


50 


8*» GO' 


175 


7.63 


174.72 


2.29 


87.37 


3° 


30' 


87.50 


8*' GO' 


10 


200 


11.11 


199.48 


2.46 


89.64 


4 


30 


110.20 


10 


20 


200 


11.11 


199.48 


2.64 


91.83 


4 


30 


108.00 


20 


30 


200 


11.11 


199.48 


2.80 


93.94 


4 


30 


105.88 


^ 30 


40 


200 


11.11 


199.48 


2.96 


95.96 


4 


30 


103.85 


40 


50 


200 


11.11 


199.48 


3.10 


97.91 


4 


30 


101.89 


50 


9? GO' 


200 


11.11 


199.48 


3.26 


99.79 


4° 


30' 


100.00 


9° GO' 


10 


225 


15.50 


224.09 


3.48 


102.03 


5 


37i 


122.73 


10 


20 


225 


15.50 


224.09 


3.69 


104.20 


5 


37i 


120.54 


20 


30 


225 


15.50 


224.09 


3.90 


106.29 


5 


37i 


118.42 


30 


40 


225 


15.50 


224.09 


4.10 


108.32 


5 


37i 


116.38 


40 


50 


225 


15.50 


224.09 


4.29 


110.28 


5 


37i 


114.41 


50 


10« GO' 


225 


15.50 


224.09 


4.48 


112.17 


5° 


37V 


112.50 


10° 00' 


10 


250 


20.91 


248 . 50 


4.74 


114.37 


6 


52i 


135.25 


10 


20 


250 


20.91 


248 . 50 


5.00 


116.53 


6 


52i 


133.06 


20 


30 


250 


20.91 


248 . 50 


5.25 


118.62 


6 


52i 


130.95 


30 


40 


250 


20.91 


248 . 50 


5.50 


120.64 


6 


52i 


128.91 


40 


50 


250 


20.91 


248.50 


5.73 


122.60 


6 


52i 


126.92 


50 


110 00' 


250 


20.91 


248.50 


5.96 


124.50 


6<' 


52i' 


125.00 


U'' 00' 



577 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 

Dimensions of various l°-per-25-feet spirals. — Part C. 

Values of AN, 





Degree of curve. 








A 




































2^ 


3° 


40 


5° 


6° 


70 


8^ 


9° 


10° 


11° 


2 


0.00 


0.00 


0.00 


0.01 


0.02 


0.03 


0.04 


0.06 


0.08 


0.11 


4 


0.00 


0.00 


0.01 


0.02 


0.03 


0.05 


0.08 


0.11 


0.16 


0.21 


6 


0.00 


0.01 


0.01 


0.03 


0.05 


0.08 


0.12 


0.17 


0.23 


0.31 


8 


0.00 


0.01 


0.02 


0.04 


0.07 


0.11 


0.16 


0.23 


0.31 


0.42 


10 


0.00 


0.01 


0.02 


0.05 


0.08 


0.13 


0.20 


0.29 


0.39 


0.52 


12 


0.00 


0.01 


0.03 


0.06 


0.10 


0.16 


0.24^ 


0.34 


0.47 


0.63 


14 


0.00 


0.01 


0.03 


0.07 


0.12 


0.19 


0.28 


0.40 


0.55 


0.73 


16 


0.00 


0.02 


0.04 


O.K)8 


0.13 


0.21 


0.32 


0.46 


0.63 


0.84 


18 


0.00 


0.02 


0.04 


0.09 


0.15 


0.24 


0.36 


0.52 


0.71 


0.94 


20 


0.00 


0.02 


0.05 


0.10 


0.17 


0.27 


0.40 


0.57 


0.79 


1.05 


22 


0.01 


0.02 


0.05 


0.11 


0.19 


0.30 


0.44 


0.63 


0.87 


1.16 


24 


0.01 


0.02 


0.06 


0.12 


0.20 


0.32 


0.49 


0.69 


0.95 


1.27 


26 


0.01 


0.03 


0.06 


0.13 


0.22 


0.35 


0.53 


0.75 


1.03 


1.38 


28 


0.01 


0.03 


0.07 


0.14 


0.24 


0.38 


0.57 


0.81 


1.11 


1.49 


30 


0.01 


0.03 


0.07 


0.15 


0.26 


0.41 


0.61 


0.87 


1.20 


1.60 


32 


0.01 


0.03 


0.08 


0.16 


0.27 


0.44 


0.65 


0.93 


1.28 


1.71 


34 


0.01 


0.03 


0.08 


0.17 


0.29 


0.47 


0.70 


1.00 


1.37 


1.82 


36 


0.01 


0.04 


0.09 


0.18 


0.31 


0.49 


0.74 


1.06 


1.45 


1.94 


38 


0.01 


0.04 


0.09 


0.19 


0.33 


0.52 


0.79 


1.12 


1.54 


2.05 


40 


0.01 


0.04 


0.10 


0.20 


0.35 


0.55 


0.83 


1.19 


1.63 


2. 17 


42 


0.01 


0.04 


0.10 


0.21 


0.37 


0.58 


0.88 


1.25 


1.72 


2.28 


44 


0.01 


0.04 


0.11 


0.22 


0.38 


0.62 


0.92 


1.32 


1.81 


2.41 


46 


0.01 


0.05 


0.12 


0.23 


0.40 


0.65 


0.97 


1.38 


1.90 


2.53 


48 


0.01 


0.05 


0.12 


0.24 


0.42 


0.68 


1.02 


1.45 


1.99 


2.65 


50 


0.01 


0.05 


0.13 


0.25 


0.44 


0.71 


1.07 


1.52 


2.09 


2.78 


52 


0.01 


0.05 


0.13 


0.27 


0.46 


0.74 


1.11 


1.59 


2.18 


2.91 


54 


0.01 


0.06 


0.14 


0.28 


0.49 


0.78 


1.16 


1.66 


2.28 


3.04 


56 


0.01 


0.06 


0.14 


0.29 


0.51 


0.81 


1.21 


1.74 


2.38 


3.17 


58 


0.02 


0.06 


0.15 


0.30 


0.53 


0.85 


1.27 


1.81 


2.48 


3.31 


60 


0.02 


0.06 


0.16 


0.31 


0.55 


0.88 


1.32 


1.88 


2.58 


3.44 


62 


0.02 


0.07 


0.16 


0.33 


0.57 


0.92 


1.37 


1.96 


2.69 


3.58 


64 


0.02 


0.07 


0.17 


0.34 


0.60 


0.95 


1.43 


2.04 


2.80 


3.73 


66 


0.02 


0.07 


0.18 


0.35 


0.62 


0.99 


1.48 


2.12 


2.91 


3.87 


68 


0.02 


0.07 


0.18 


0.37 


0.64 


1.03 


1.54 


2.20 


3.02 


4.02 


70 


0.02 


0.08 


0.19 


0.38 


0.67 


1.07 


1.60 


2.28 


3.13 


4.18 


72 


0.02 


0.08 


0.20 


0.40 


0.69 


1.11 


1.66 


2.37 


3.25 


4.33 


74 


0.02 


0.08 


0.20 


0.41 


0.72 


1.15 


1.72 


2.46 


3.38 


4.49 


76 


0.02 


0.09 


0.21 


0.43 


0.74 


1.19 


1.79 


2.55 


3.50 


4.66 


78 


0.02 


0.09 


0.22 


0.44 


0.77 


1.23 


1.85 


2.64 


3.63 


4.83 


80 


0.02 


0.09 


0.23 


0.46 


0.80 


1.28 


1.92 


2.74 


3.76 


5.00 


82 


0.02 


0.09 


0.24 


0.47 


0.83 


1.32 


1.99 


2.83 


3.89 


5.18 


84 


0.03 


0.10 


0.24 


0.49 


0.86 


1.37 


2.06 


2.94 


4.03 


5.37 


86 


0.03 


0.10 


0.25 


0.51 


0.89 


1.42 


2.13 


3.04 


4.18 


5.56 


88 


0.03 


0.11 


0.26 


0.53 


0.92 


1.47 


2.21 


3.15 


4.33 


5.76 


90 


0.03 


0.11 


0.27 


0.55 


0.95 


1.52 


2.29 


3.26 


4.48 


5.96 


92 


0.03 


0.11 


0.28 


0.56 


0.99 


1.58 


2.37 


3.38 


4.64 


6.17 


94 


0.03 


0.12 


0.29 


0.58 


1.02 


1.63 


2.45 


3.50 


4.80 


6.39 


96 


0.03 


0.12 


0.30 


0.61 


1.06 


1.69 


2.54 


3.62 


4.97 


6.62 


98 


0.03 


0.13 


0.31 


0.63 


1.10 


1.75 


2.63 


3.75 


5.15 


6.86 


100 


0.03 


0.13 


0.32 


0.65 


1.13 


1.82 


2.72 


3.89 


5.34 


7.11 



■578 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 



c 
"o 


rHCqcC^lOCDC^OOOiO 


^ 


OCDC«-iCOt*050C<JCD^ 

oa>t^rHcomc»CNii>.co 
oa>0503t>^coa>coo 


iDos^cs^cncot^i— c* 

t-l I— 1 i-l 1— 1 <M CM 


5$ 
faC 

.3 


I'^IO OOICOICM 05lt> O. t^ CO 
C»-t^OOCDCOC^OCCOOi— 1 

coeoooi— (t^oscoti^oOi-H 

Ot-r-HlOD-OSr-ICO^CD 


OSOSOOOOi— Ir-Hi-Hi— 1 


8 


OSlOt^i— lCNJCOC0O5CMt>- 

o^c<jc>.03OO03mr— i 
i-iiniocMososcsiot-co 


OOrHCOlOCDlOCSIOr-l 


-e- 

> 

bo 
o 


CD 00 t>^ OilCM 05 CQIOi OICO 
CDCOOOCOCSIOOOOCDLOO 

O'y^cDOCNcocqajr-ico 
ooeoeoooeocqi>.oocoir5 

lOlOrHlOOiCvJriHCOeOO 


iococ^t>.t*oooooocoa> 


-0- 

i 

—1 


100 IDIOI-"* t-ICDIOIOInH CO 
O300-<:t<CO<MCDO5CMCDO5 

osojoscocDcqccooiot:^ 

Od 03 03 O) 03 03 00 t^ CD ^ 
a>03CX)03030a05CD05C33 


CDO3C»00O5C»C»a>a.O5 


-0- 

ho 


■>* cq oicj) oicoicx 00 -^IG 
ooo^oocN^^-cDCDo^oO'* 

OC^OOOIOOCOCJXM-^ 
•"^HnHrHrJlrHCDOOOOOOCD 

05-^t>-cj)i-icqco"«d<incD 


C^-OOOOOOOiOiOiOSOiOi 


■e- 

i 


IC5ICOICO (MI'«;J1ICNI OOIO 05 O 
OSCDOOCOnHOOOrHCOt^ 
0503050S0500C^«OCSJOO 
CnO3O3a>O5C»C»O3C»00 




-e- 


lb- CqiCOIi-HllOKM 05 O t>lt> 

oocDcqt>-ocqt-ic»cqrH 
ocqinoocooo^oooco 
OOOOrH,-(CqcoeO">;i< 




Total 
Central Angle 


oooooooooo 
cocoooeocooocoeo 

o 

OiHCOWt^O^OOCMt^ 
iH i-H i-H cq CSI 


'I 


fH cq CO -^i* lO CO C* 00 OJ o 



H 


o 


« 


a 


< 


J3 


PU 


c 




c 




o 


C3 


4^ 




o 


A 


a 





"lb 
CI 


cq 

•<* 


o 

o 


o 


CM 


CD 

o 


o 

CO 


§ 


o 

CO 


§ 


s 


c 


CO 

in 


o 
in 


00 
CO 


in 




o 
o 


o 


in 

o 


in 


o 

CO 


o 
o 




1— 1 


CO 
i-H 


in 

I-) 


.— 1 


cq 

r-i 


I-I 


05 


c- 


Ti< 


CM O 




o 

CO 


CM 

m 


in 

1—1 


03 
CO 


in 

o 


o 

CO 


o 

o 


o 

CO 


o 
o 


o 
o 


§ 


05 


in 

CO 


o 

CO 


o 

CO 


r-i 


in 


o 


g 


CQ 


in 


o 

o 


o 

CO 




I-I 


CO 


cq 
1— 1 


I-i 
I-I 


Oi 


00 


CD 


■<^ 


CN 


o 


<M 




"in 


o 

1— 1 


CO 
CO 


s 


o 

CO 


o 
o 


o 

CO 


o 


O 

o 


o o 

O CO 


QC 


CO 


in 


CM 


o 

CO 


o 


in 

CO 


CM 

in 


§ 


§ 


in t- 

r-l CO 




o 

pH 
r-l 


o 

r-H 


Oi 


CO 


l>. 


in 


CO 


CM 


o 


cq ^ 




o 


CO 


CQ 

o 


o 

CO 


o 

o 


o 

CO 


o 
o 


O 
O 


o 
o 


o o 
CO o 


r^ 


o 

o 


CO 

1— 1 


in 


o 


o 
m 


CM 
CM 


in 


o 

o 


§ 


b- in 
o cq 




Oi 


00 


r> 


CD 


■«* 


CO 




o 


CM 


-1* CD 




o 

CO 


o 


o 

CO 


o 
o 


o 

CO 


c 
c 


c 
c 


o 
o 


o 

CO 


O O 
O CO 


o 




o 

o 


o 


in 

o 


CNJ 

m 


o 

CO 


c 

c 


in 


00 


§s 




CO 


QD 


in 


^ 


cq 


r-l 


o 


rH 


CO 


in 


t^ 



1 
i 


O 

o 


o 

CO 


o 
o 


o 

CO 


o 
o 


o 
o 


o 
o 


o 

CO 


o 

o 


CO in 


i iC 


"in t^ o cq 

■«^ O CM CM 


in 


o 
o 


o 

CO 


o 


in 

in 


cq 05 
in in 


1 


■^ ^ CO CM 


i— I 


o 


1—1 


CO 


■«^ 


CD 00 




o o o o 

CO O CO o 


§ 


o 
o 


o 

CO 


§ 


o 

CO 


in 00 

in rH 


r}< 


V. in CM o 
o CO in o 


o 
o 


in 

1— 1 


CO 


o 

I-i 


cq 
in 


^ t- 
^ "^ 




00 cq 1— 1 1— 1 


o 


r-l 


cq 


'dH 


in 


!>. 05 




o o o 

O CO o 


o 
o 


o o 

O CO 


o 
o 


o 

CO 


CD 

m 


r-l O 

cq Ti4 


CO 


o CM m 
m csi ^ 


§ 


8£ 


m 
cq 


cq 
in 


Oi 
cq 


b- -^ 

I— 1 I-I 




r-i r-i <D 


o 


,-i eq 


CO 


'ilH 


CD 


00 o 

1— 1 




CO O 


o 

o 


§ 


o o 

CO o 


o 

CO 


00 

in 


cq 


in o 

^ o 


W 


CM O 

in CO 


o 
o 


in 


o- o 

CO '^ii 


cq 
in 




5 


a> cq 
cq cq 




'^oo 


o 


o 


I-I cq 


CO 


in 


CD 


00 o 

pH 




o 
o 


o 
o 


o o 

O CO 


o o 

O CO 


03 

m 


CD 

cq 


O 

in 


00 bo 

O i-H 


iH 


in 


§ 


o c«- 

CO o 


in cq 
in in 


a> 
m 


1— 1 


Til 


cq « 
cq o 




o 


o 


O r-i 


rH CM 


CO 


to 


CD 


CO o 

I-I 




o 
o 


o o o 

O CO o 


o o 

CO o 


00 

cq 


CO 

in 


in 

I— 1 


o in 

CO CO 


C^ 


§ 


in t>. o 

nH CO r-l 


CM in 

in *;*< 


b- 


03 

in 


cq 

CM 


^ CD 

in CO 




o 


O O I-I 


.-1 CM 


CO 


-^ 


CO 


C^ Oi 



O^THWW'*tOCDi>-a005© 



579 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 
Dimensions of various 2°-per-25-feet spirals. — Part B. 



^of- 


L'gth 
of 


ZK 


QK 


A'N 


NQ 


A' to Z. 


"^ 








curve. 


spiral 










Defl. 


Dist. 


curve. 


4° 00' 


25 


0.11 


25.00 


0.05 


12.50 


0° 


15' 


12.50 


4° 00' 


20 


50 


0.55 


50.00 


0.09 


15.38 





45 


34.62 


20 


40 


50 


0.55 


50.00 


0.12 


17.85 





45 


32.14 


40 


5° 00' 


50 


0.55 


50.00 


0.15 


19.99 





45 


30.00 


5° 00' 


20 


50 


0.55 


50.00 


0.18 


21.86 





45 


28.13 


20 


40 


50 


0.55 


50.00 


0.20 


23.52 





45 


26.47 


40 


6° 00' 


50 


0.55 


50.00 


0.22 


24.99 


0° 


45' 


25.00 


6° 00' 


20 


75 


1.52 


74.98 


0.29 


27.61 




30 


47.37 


20 


40 


75 


1.52 


74.98 


0.35 


29.97 




30 


45.00 


40 


f? 00' 


75 


1.52 


74.98 


0.40 


32.11 




30 


42.86 


7° 00' 


20 


75 


1.52 


74.98 


0.46 


34.06 




30 


40.91 


20 


40 


75 


1.52 


74.98 


0.50 


35.84 




30 


39.13 


40 


8° 00' 


75 


1.52 


74.98 


0.54 


37.46 


1° 


30' 


37.50 


8° 00' 


20 


100 


3.27 


99.92 


0.65 


39.94 


2 


30 


60.00 


20 


40 


100 


3.27 


99.92 


0.75 


42.24 


2 


30 


57.69 


40 


9^ 00' 


100 


3.27 


99.92 


0.85 


44.37 


2 


30 


55.56 


9° OO' 


20 


100 


3.27 


99.92 


0.93 


46.35 


2 


30 


53.57 


20 


40 


100 


3.27 


99.92 


1.01 


48.20 


2 


30 


51.72 


40 


10° 00' 


100 


3.27 


99.92 


1.09 


49.92 


2° 


30' 


50.00 


10° 00' 


20 


125 


5.99 


124.77 


1.24 


52.30 


3 


45 


72.58 


20 


40 


125 


5.99 


124.77 


1.39 


54.55 


3 


45 


70.31 


40 


IP 00' 


125 


5.99 


124.77 


1.53 


56.68 


3 


45 


68.18 


11° 00' 


20 


125 


5.99 


124.77 


1.66 


58.67 


3 


45 


66.18 


20 


40 


125 


5.99 


124.77 


1.78 


60.55 


3 


45 


64.29 


40 


12° 00' 


125 


5.99 


124.77 


1.90 


62.33 


3° 


45' 


62.50 


12° 00' 


20 


150 


9.90 


149.46 


2.11 


64.68 


5 


15 


85.14 


20 


40 


150 


9.90 


149.46 


2.31 


66.86 


5 


15 


82.89 


40 


13° 00' 


150 


9.90 


149.46 


2.51 


68.97 


5 


15 


80,77 


13° 00' 


20 


150 


9.90 


149.46 


2.69 


70.97 


5 


15 


78.75 


20 


40 


150 


9.90 


149.46 


2.87 


72.88 


5 


15 


76.83 


40 


14° 00' 


150 


9.90 


149.46 


3.03 


74.69 


5» 


15' 


75.00 


14° 00' 


20 


175 


15.21 


173.89 


3.30 


76.93 


7 


00 


97.67 


20 


40 


175 


15.21 


173.89 


3.57 


79.12 


7 


00 


95.45 


40 


15° 00' 


175 


15.21 


173.89 


3.83 


81.22 


7 


00 


93.33 


15° 00' 


20 


175 


15.21 


173.89 


4.08 


83.22 


7 


00 


91.30 


20 


40 


175 


15.21 


173.89 


4.31 


85.14 


7 


00 


89.36 


40 


16° 00' 


175 


15.21 


173.89 


4.54 


86.98 


7° 


00' 


87.50 


16° 00' 


20 


200 


22.10 


197.92 


4.87 


89.15 


9 


00 


110.20 


20 


40 


200 


22.10 


197.92 


5.21 


91.31 


9 


00 


108.00 


40 


17° 00' 


200 


22.10 


197.92 


5.54 


93.39 


9 


00 


105.88 


17° 00' 


20 


200 


22.10 


197.92 


5.86 


95.38 


9 


00 


103.85 


20 


40 


200 


22.10 


197.92 


6.16 


97.30 


9 


00 


101.89 


40 


18° 00' 


200 


22.10 


197.92 


6.45 


99.15 


9° 


00' 


100.00 


18° 00' 


20 


225 


30.75 


221.38 


6.86 


101.27 


11 


15 


122.73 


20 


40 


225 


30.75 


221.38 


7.28 


103.39 


11 


15 


120.54 


40 


19° 00' 


225 


30.75 


221.38 


7.69 


105.45 


11 


15 


118.42 


19° 00' 


20 


225 


30.75 


221.38 


8.09 


107.43 


11 


15 


116.38 


20 


40 


225 


30.75 


221.38 


8.47 


109.34 


11 


15 


114.41 


40 


20° 00' 


225 


30.75 


221.38 


8.83 


111.19 


11° 


15' 


112.50 


20° 00' 


20 


250 


41.32 


244.03 


9.31 


113.23 


13 


45 


135.25 


20 


40 


250 


41.32 


244.03 


9.82 


115.32 


13 


45 


133.06 


40 


21° 00' 


250 


41.32 


244.03 


10.32 


117.35 


13 


45 


130.95 


21° 00' 


20 


250 


41.32 


244.03 


10.80 


119.30 


13 


45 


128.91 


20 


40 


250 


41.32 


244.03 


11.26 


121.20 


13 


45 


126.92 


40 


22° 00' 


250 


41.32 


244.03 


11.71 


123.04 


13° 


45' 


125.00 


22° 00' 



680 



TABLE IV.— ELEMENTS OF TRANSITION CURVES. 

Dimensions of various 2°-per-25-feet spirals. — Part C, 

Values of AN, 







Degree of curve. 




A 
































4° 


6° 


8'' 


10° 


12° 


14° 


16° 


18° 


20° 


22° 


40 


0.00 


0.01 


0.02 


0.04 


0.07 


0.11 


0.18 


0.23 


0.31 


0.41 


6 


0.00 


0.01 


0.03 


0.06 


0.10 


0.16 


0.24 


0.34 


0.46 


0.61 


8 


0.00 


0.02 


0.04 


0.08 


0.13 


0,21 


0.32 


0.45 


0.62 


0.82 


10 


0.00 


0.02 


0.05 


0.10 


0.17 


0.27 


0.40 


0.56 


0.77 


1.02 


12 


0.01 


0.02 


0.06 


0.11 


0.20 


0.32 


0.48 


0.68 


0.93 


1.23 


14 


0.01 


0.03 


0.07 


0.13 


0.23 


0.37 


0.56 


0.79 


1.08 


1.44 


16 


0.01 


0.03 


0.08 


0.15 


0.27 


0.43 


0.64 


0.91 


1.24 


1.65 


18 


0.01 


0.03 


0.09 


0.17 


0.30 


0.48 


0.72 


1.02 


1.40 


1.85 


20 


0.01 


0.04 


0.10 


0.19 


0.34 


0.53 


0.80 


1.14 


1.56 


2.06 


22 


0.01 


0.04 


0.11 


0.21 


0.37 


0.59 


0.88 


1.25 


1.72 


2.28 


24 


0.01 


0.05 


0.12 


0.23 


0.40 


0.64 


0.96 


1.37 


1.88 


2.49 


26 


0.01 


0.05 


0.13 


0.25 


0.44 


0.70 


1.05 


1.49 


2.04 


2.70 


28 


0.01 


0.05 


0.14 


0.27 


0.47 


0.76 


1.13 


1.61 


2.20 


2.92 


30 


0.01 


0.06 


0.15 


0.29 


0.51 


0.81 


1.22 


1.73 


2.37^ 


3.14 


32 


0.02 


0.06 


0.16 


0.31 


0.54 


0.87 


1.30 


1.85 


2.53 


3.36 


34 


0.02 


0.07 


0.17 


0.33 


0.58 


0.93 


1.39 


1.97 


2.70 


3.58 


36 


0.02 


0.07 


0.18 


0.35 


0.62 


0.99 


1.47 


2.10 


2.87 


3.80 


38 


0.02 


0.08 


0.19 


0.37 


0.65 


1.04 


1.56 


2.22 


3.04 


4.03 


40 


0.02 


0.08 


0.20 


0.40 


0.69 


1.10 


1.65 


2.35 


3.22 


4.26 


42 


0.02 


0.08 


0.21 


0.42 


0.73 


1.16 


1.74 


2.48 


3.39 


4.49 


44 


0.02 


0.09 


0.22 


0.44 


0.77 


1.23 


1.83 


2.61 


3.57 


4.73 


46 


0.02 


0.09 


0.23 


0.46 


0.81 


1.29 


1,93 


2.74 


3.75 


4.97 


48 


0.02 


0.10 


0.24 


0.48 


0.85 


1.35 


2.02 


2.87 


3.93 


5.21 


50 


0.03 


0.10 


0.25 


0.51 


0.89 


1.41 


2.11 


3.01 


4.12 


5.46 


52 


0.03 


0.11 


0.27 


0.53 


0.93 


1.48 


2.21 


3.15 


4.31 


5.71 


54 


0.03 


0.11 


0.28 


0.55 


0.97 


1.54 


2.31 


3.29 


4.50 


5.97 


56 


0.03 


0.12 


0.29 


0.58 


1.01 


1.61 


2.41 


3.43 


4.70 


6.23 


58 


0.03 


0.12 


0.30 


0.60 


1.05 


1.68 


2.51 


3.58 


4.90 


6.49 


60 


0.03 


0.13 


0.31 


0.63 


1.10 


1.75 


2.62 


3.73 


5.10 


6.76 


62 


0.03 


0.13 


0.33 


0.65 


1.14 


1.82 


2.73 


3.88 


5.31 


7.04 


64 


0.03 


0.14 


0.34 


0.68 


1.19 


1.90 


2.83 


4.03 


5.52 


7.32 


66 


0.03 


0.14 


0.35 


0.71 


1.23 


1.97 


2.95 


4.19 


5.74 


7.60 


68 


0.04 


0.15 


0.37 


0.73 


1.28 


2.05 


3.06 


4.35 


5.96 


7.90 


70 


0.04 


0.15 


0.38 


0.76 


1.33 


2.12 


3.18 


4.52 


6.19 


8.20 


72 


0.04 


0.16 


0.40 


0.78 


1.38 


2.20 


3.30 


4.69 


6.42 


8.51 


74 


0.04 


0.16 


0.41 


0.81 


1.43 


2.28 


3.42 


4.86 


6.66 


8.82 


76 


0.04 


0.17 


0.43 


0.84 


1.48 


2.37 


3.54 


5.04 


6.90 


9.15 


78- 


0.04 


0.18 


0.44 


0.88 


1.54 


2.46 


3.67 


5.22 


7.15 


9.48 


80 


0.05 


0.18 


0.46 


0.91 


1.59 


2.54 


3.81 


5.41 


7.41 


9.82 


82 


0.05 


0.19 


0.47 


0.94 


1.65 


2.64 


3.94 


5.61 


7.68 


10.18 


84 


0.05 


0.20 


0.49 


0.98 


1.71 


2.73 


4.08 


5.81 


7.95 


10.54 


86 


0.05 


0.20 


0.51 


1.02 


1.77 


2.83 


4.23 


6.02 


8.24 


10.92 


88 


0.05 


0.21 


0.53 


1.05 


1.83 


2.93 


4.38 


6.23 


8.53 


11.31 


90 


0.05 


0.22 


0.55 


1.09 


1.90 


3.03 


4.54 


6.45 


8.83 


11.71 


92 


0.06 


0.23 


0.56 


1.13 


1.97 


3.14 


4.70 


6.68 


9.15 


12.12 


94 


0.06 


0.23 


0.58 


1.17 


2.04 


3.25 


4.86 


6.92 


9.47 


12.56 


96 


0.06 


0.24 


0.61 


1.21 


2.11 


3.37 


5.04 


7.17 


9.81 


13.00 


98 


0.06 


0.25 


0.63 


1.25 


2.19 


3.49 


5.22 


7.42 


10.16 


13.47 


100 


0.06 


0.26 


0.65 


1.30 


2.26 


3.61 


5.41 


7.69 


10.53 


13.95 



581 







TABLE V 


.^-LOGARITHMS OF NUMBERy. 














- 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


100 


00 000 


043 


087 


130 


173 


216 


260 


303 


346 


389 


























^ 43 43 42 41 


101 


432 


475 


518 


561 


604 


646 


689 


732 


775 


817 


.1 


4.3 


4.3 


4.2 


4.1 


102 


860 


902 


945 


987 


*030 


*072 


ni4 


*157 


*199 


*241 


.2 


8 


• 7 


8.6 


84 


8.2 


103 


01 283 


326 


368 


410 


452 


494 


536 


578 


619 


661 


.3 


13 


.0 


12.9 


12.6 


12.3 


104 


703 


745 


787 


828 


870 


911 


953 


994 


*036 


*077 


• 4 


17 


•4 


17.2 


16.8 


16.4 


105 


02 119 


160 


201 


243 


284 


325 


366 


407 


448 


489 


• 5 


21 


.7 


21.5 


21.0 


20.5 


106 


530 


571 


612 


653 


694 


735 


775 


816 


857 


898 


• 6 


26 


.1 


25.8 


25.2 


24.6 


107 


938 


979 


*019 


*060 


*100 


*141 


*181 


*221 


*262 


*302 


.7 


30 


.4 


30.] 


29.4 


28.7 


108 


03 342 


382 


422 


463 


503 


543 


583 


,623 


663 


703 


8 


34 


8 


34.4 


33.6 


32.8 


109 


742 
04 139 


782 


822 


862 
257 


901 
297 


941 
336 


981 
375 


*020 


^^060 


*100 
493 


.9 


39 


.1 


38.7 


37.8 


36-9 


110 


178 


218 


415 


454 


40 40 39 38 
























111 


532 


571 


610 


649 


688 


727 


766 


805 


844 


883 


• 1 


4.0 


4.0 


3.9 


38 


112 


922 


960 


999 


*038 


*076 


*115 


*154 


*192 


*231 


*269 


.2 


8.1 


8 


.0 


7 


.8 


7.6 


113 


05 308 


346 


384 


423 


461 


499 


538 


576 


614 


652 


• 3 


12.1 


12 


• 


n 


.7 


11.4 


114 


690 


728 


766 


804 


842 


880 


918 


956 


994 


*032 


• 4 


16.2 


16 


.0 


15 


.6 


15.2 


115 


06 070 


107 


145 


183 


220 


258 


296 


333 


371 


408 


• 5 


20.2 


20 


• 


19 


• 5 


19.0 


116 


446 


483 


520 


558 


595 


632 


670 


707 


744 


781 


.6 


24.3 


24 


• 


23 


.4 


22.8 


117 


818 


855 


893 


930 


967 


*004 


*040 


*077 


*114 


n5i 


.7 


28.3 


28 


.0 


27 


• 3 


26.6 


118 


07 188 


225 


261 


298 


335 


372 


408 


445 


481 


518 


8 


32.4 


32 





31 


.2 


30.4 


119 


554 
918 


591 


627 


664 
*026 


700 
*062 


737 
*098 


773 


809 


845 


882 


• 9 


36.4 


36 





35 


•1 


34.2 


120 


954 


990 


*134 


*170 


*206 


*242 


37 37 ; 36 35 
























121 


08 278 


314 


350 


386 


422 


457 


493 


529 


564 


600 


•r 


3.7 


3.7 


3.6 


3.5 


122 


636 


671 


707 


742 


778 


813 


849 


884 


920 


955 


.2 


7.5 


7 


4 


7 


2 


7.0 


123 


990 


*026 


*061 


*096 


*131 


*166 


*202 


*237 


*272 


*307 


• 3 


11.2 


11 


1 


10 


8 


10.5 


124 


09 342 


377 


412 


447 


482 


517 


552 


586 


621 


656 


•4 


15.0 


14 


8 


14 


4 


14.0 


125 


691 


725 


760 


795 


830 


864 


899 


933 


968 


*002 


5 


18.7 


18 


5 


18 





17.5 


126 


10 037 


071 


106 


140 


174 


209 


243 


277 


312 


346 


• 6 


22.5 


22 


2 


21 


6 


21.0 


127 


380 


414 


448 


483 


517 


551 


585 


619 


653 


687 


•7 


26.2 


25 


9 


25 


2 


24.5 


128 


721 


755 


789 


822 


856 


890 


924 


958 


991 


*025 


• 8 


30.0 


29 


6 


28 


8 


28.0 


129 


11 059 
394 


092 


126 


160 
494 


193 


227 
561 


260 


294 
627 


327 


361 
694 


9 


33.7 


33 


3 


32 


4 


31.5 


130 


427 


461 


528 


594 


661 


f 
























3l 34 33 32 
























131 


727 


760 


793 


826 


859 


892 


925 


958 


991 


*024 


.1 


3.4 


3.4 


33 


3.2 


132 


12 057 


090 


123 


156 


189 


221 


254 


287 


320 


352 


.2 


6 


g 


6. 


8 


6 


6 


6 


4 


:v33 


385 


418 


450 


483 


515 


548 


580 


613 


645 


678 


3 


10 


3 


10. 


2 


9 


9 


9 


6 


134 


710 


743 


775 


807 


840 


872 


904 


937 


969 


*001 


4 


13 


8 


13 


6 


13 


2 


12 


8 


135 


13 033 


065 


097 


130 


162 


194 


226 


258 


290 


322 


.5 


17 


2 


17 





16 


5 


16 





136 


354 


386 


417 


449 


481 


513 


545 


577 


608 


640 


• 6 


20 


7 


20. 


4 


19 


8 


19 


2 


137 


672 


703 


735 


767 


798 


830 


862 


893 


925 


956 


• 7 


24 


1 


23. 


8 


23 


1 


22 


4 


138 


988 


*019 


*051 


*082 


*113 


*145 


*176 


*207 


*239 


*270 


8 


27 


g 


27. 


2 


26 


4 


25 


6 


139 


14 301 


332 


364 


395 


426 


457 


488 


519 


550 


582 


• 9 


31 


b 


30. 


6 


29 


7 


28 


8 


140 


613 


644 


675 


706 


736 


767 


798 


829 


860 


891 


31 . 31 . 30 . 29 
























141 


922 


955 


983 


*014 


*045 


*075 


*106 


*137 


*167 


*198 


.1 


3 11 


3.1 


3.0 


2.9 


142 


15 229 


259 


290 


320 


351 


381 


412 


442 


473 


503 


.2 


6 


i 


6-2 


6 





5.8 


143 


533 


564 


594 


624 


655 


685 


J15 


745 


776 


806 


.3 


9 


i. 


9.3 


9 





8.7 


144 


836 


866 


896 


926 


956 


987 


*017 


*047 


*077 


*107 


• 4 


12 


6 


12.4 


12 





11.6 


145 


16 137 


166 


196 


226 


256 


286 


316 


346 


376 


405 


.5 


15 


7 


15.5 


15 





14.5 


146 


435 


465 


494 


524 


554 


584 


613 


643 


672 


702 


.6 


18 


9 


18. 6 


18. 





17.4 


147 


731 


761 


791 


820 


849 


879 


908 


938 


967 


997 


• 7 


22 





21.7 


21. 





20.3 


148 


17 026 


055 


085 


114 


143 


172 


202 


231 


260 


289 


.8 


25 


2 


24.8 


24. 





23.2 


149 


318 


348 


377 


406 


435 


464 


493 


522 


551 


580 


.9 


28 


3 


27.9 


27.01 


26.1 


160 


609 


638 


667 
2 


696 
3 


725 
4 


75§ 
5 


782 
6 


811 

7 


840 


869 




■ 


N. 





1 


8 


9 








P. 


P 


• 







582 









TABLE y 


'.—LOGARITHMS 


OF NUMBERS 


,. 












N. 





1 


2 


3 

696 


4 

725 


5 

753 


6 

782 


7 
811 


8 
840 


9 


P. P. 


150 


17 609 


638 


667 


869 


39 38 37 
























151 


897 


926 


955 


984 


*012 


*041 


^070 


*098 


=^127 


*156 


.1 


2.9 


2.8 


2.7 


152 


18 184 


213 


241 


270 


298 


327 


355 


384 


412 


440 


.2 


t 


.8 


5 


.6 


5 


4 


153 


469 


497 


526 


554 


582 


611 


639 


667 


695 


724 


.3 


I 


• 7 


i 


.4 


8 


1 


154 


752 


780 


808 


836 


864 


893 


921 


949 


977 


*005 


.4 


1] 


.6 


1] 


.2 


10 


8 


155 


19 033 


061 


089 


117 


145 


173 


201 


229 


256 


284 


.5 


14 


• 5 


1^ 


.0 


13 


5 


156 


312 


340 


368 


396 


423 


451 


479 


507 


534 


562 


.6 


17 


•4 


16 


.8 


16 


2 


157 


590 


617 


645 


673 


700 


728 


755 


,783 


810 


838 


.7 


2C 


• 3 


19 


.6 


18 


9 


158 


865 


893 


920 


948 


975 


^003 


=^030 


*057 


=^085 


ni2 


.8 


23 


• 2 


22 


.4 


21 


6 


159 


20 139 


167 


194 


221 


249 


276 


303 


330 


357 


385 


.9 


26 


.1 


25 


.2 


24 


3 


160 


412 


439 


466 


493 


520 


547 


574 


601 


628 


655 




161 


682 


709 


736 


763 


790 


817 


844 


871 


898 


924 


.1 


/CO 

2.61 


,«o 

2.6 


162 


951 


978 


*005 


^032 


^058 


*085 


=^112 


^139 


^165 


*192 


• 2 


5 


• 3 


5 


.2 


163 


21 219 


245 


272 


298 


325 


352 


378 


405 


431 


458 


.3 


7 


. 9 


7 


8 


164 


484 


511 


537 


564 


590 


616 


643 


669 


695 


722 


.4 


10 


. 6 


10 


4 


165 


748 


774 


801 


827 


853 


880 


906 


932 


958 


984 


.5 


13 


• 2 


13 


.0 


166 


22 Oil 


037 


063 


089 


115 


141 


167 


193 


219 


245 


• 6 


15 


.9 


15 


.6 


167 


271 


297 


323 


349 


375 


401 


427 


453 


479 


505 


.7 


18 


.5 


18 


.2 


168 


531 


557 


582 


608 


634 


660 


686 


711 


737 


763 


.8 


21 


.2 


20 


8 


169 


788 


814 


840 


865 


891 


917 


942 


968 


994 


*019 


.9 


23 


.8 


23 


4 


170 


23 045 


070 


096 


121 


147 


172 


198 


223 


249 


274 


35 35 34 
























171 


299 


325 


350 


375 


401 


426 


451 


477 


502 


527 


1 


2-5 


2.5 


2.4 


172 


553 


578 


603 


628 


653 


679 


704 


729 


754 


779 


2 


5 


•1 


5 


.0 


4.8 


173 


804 


829 


855 


880 


905 


930 


955 


980 


^005 


*030 


3 


7 


■ 6 


7 


.5 


7-2 


174 


24 055 


080 


105 


129 


154 


179 


204 


229 


254 


279 


4 


10 


. 2 


10 


.0 


9.6 


175 


304 


328 


353 


378 


403 


427 


452 


477 


502 


526 


5 


12 


. 7 


12 


.5 


12.0 


176 


551 


576 


600 


625 


650 


674 


699 


723 


748 


773 


6 


15 


• 3 


15 


.0 


14.4 


177 


797 


822 


846 


871 


895 


920 


944 


968 


993 


*017 


7 


17 


.8 


17 


.5 


16-8 


178 


25 042 


066 


091 


115 


139 


164 


188 


212 


237 


261 


8 


20 


.4 


20 


.0 


19-2 


179 


285 


309 


334 


358 


382 


406 


430 


455 


479 


503 


9 


22 


.9 


22 


.5 


21.6 


180 


527 


55l 


575 


599 


623 


647 


672 


696 


720 


744 


33 33 
























181 


768 


792 


816 


840 


863 


887 


911 


935 


959 


983 


.1 


2.31 


2.3 


182 


26 007 


031 


055 


078 


102 


126 


150 


174 


197 


221 


.2 


4 


7 


4 


6 


183 


245 


269 


292 


316 


340 


363 


387 


411 


434 


458 


.3 


7 





6 


9 


184 


482 


505 


529 


552 


576 


599 


623 


646 


670 


693 


.4 


9 


4 


9 


2 


185 


717 


740 


764 


787 


811 


834 


858 


881 


904 


928 


.5 


11 


7 


11 


5 


186 


951 


974 


998 


^021 


=*=044 


=^068 


^091 


ni4 


n37 


*161 


.6 


14 


1 


13 


8 


187 


27 184 


207 


230 


254 


277 


300 


323 


346 


369 


392 


.7 


16 


4 


16 


1 


188 


416 


439 


462 


485 


508 


531 


554 


577 


600 


623 


.8 


18 


8 


18 


4 


189 


646 


669 


692 


715 


738 


761 


784 


806 


829 


852 


.9 


21.11 


20.7 


190 


875 


898 


921 


944 


966 


989 


*012 


*035 


*058 


*080 


33 33 31 






191 


28 103 


126 


149 


171 


194 


217 


239 


262 


285 


307 


1 


2.2 


2-2 


2.1 


192 


330 


352 


375 


398 


420 


443 


465 


488 


510 


533 


2 


4.5 


4 


•4 


4.3 


193 


555 


578 


600 


623 


645 


668 


690 


713 


735 


758 


3 


6.7 


6 


.6 


6-4 


194 


780 


802 


825 


847 


869 


892 


914 


936 


959 


981 


4 


9.0 


8 


■ 8 


8.6 


195 


29 003 


025 


048 


070 


092 


114 


137 


159 


181 


203 


5 


11.2 


11 


.0 


10.7 


196 


225 


248 


270 


292 


314 


336 


358 


380 


402 


424 


6 


13.5 


13 


.2 


12.9 


197 


446 


468 


490 


512 


534 


556 


578 


600 


622 


644 


7 


15.7 


15 


•4 


15-0 


198 


666 


688 


710 


732 


754 


776 


798 


820 


841 


863 


8 


18.0 


17 


.6 


17.2 


199 


885 
30 103 


907 
124 


929 

146 


950 


972 


994 


*016 


*038 


*059 


*081 


9 


20.2 


19 


.8. 


19.3 


300 


168 


190 
4 


211 
5 


233 
6 


254 

7 


276 


298 




N. 





1 


2 


3 


8 


9 






P 


. I 


» 







583 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 
124 


2 

146 


3 

168 


4 

190 


5 

211 


6 

233 


7 
254 


8 
276 


9 


P. P. 


200 


30 103 


298 


23 21 
























201 


319 


341 


363 


384 


406 


427 


449 


470 


492 


513 


.1 2.2 


2.1 


202 


535 


556 


578 


599 


621 


642 


664 


685 


707 


728 


.2 4.4 


4.2 


203 


749 


771 


792 


813 


835 


856 


878 


899 


920 


941 


.3 6.6 


6.3 


204 


963 


984 


*005 


*027 


*048 


*069 


*090 


*112 


*133 


*154 


.4 8-8 


8.4 


205 


31 175 


196 


217 


239 


260 


281 


302 


323 


344 


365 


.511.0 


10.5 


206 


386 


408 


429 


450 


471 


492 


513 


534 


555 


576 


.613.2 


12.6 


207 


597 


618 


639 


660 


681 


702 


722 


743 


764 


785 


.7 15-4 


14.7 


208 


806 


827 


848 


869 


890 


910 


931 


952 


973 


994 


.8 17-6 


16-8 


209 


32 014 


035 


056 


077 


097 


118 


139 


160 


180 


201 


.9 19.8 


18.9 


310 


222 


242 


263 


284 


304 


325 


346 


366 


387 


407 


20 20 
























211 


428 


449 


469 


490 


510 


531 


551 


572 


592 


6ia 


.1 2.0 


2.0 


212 


633 


654 


674 


695 


715 


736 


756 


776 


797 


817 


.2 4.1 


4.0 


213 


838 


858 


878 


899 


919 


940 


960 


980 


*001 


*021 


.3 6.1 


6.0 


214 


33 041 


061 


082 


102 


122 


142 


163 


183 


203 


223 


.4 8.2 


8.0 


215 


244 


264 


284 


304 


324 


344 


365 


385 


405 


425 


.5 10.2 


10.0 


216 


445 


465 


485 


505 


525 


546 


566 


586 


606 


626 


.6 12.3 


12.0 


217 


646 


666 


686 


706 


726 


746 


766 


786 


806 


825 


.7 14.3 


14.0 


218 


845 


865 


885 


905 


925 


945 


965 


985 


*004 


*024 


.8 16.4 


16-0 


219 


34 044 
242 


064 
262 


084 


104 
301 


123 
321 


143 
341 


163 


183 
380 


203 
400 


222 


.9 18.4 


18.0 


320 


281 


360 


419 




221 


439 


459 


478 


498 


518 


537 


557 


576 


596 


615 


19 ""^ 

.1 1.9 


1.9 


222 


635 


655 


674 


694 


713 


733 


752 


772 


791 


811 


.2 3.< 


38 


223 


830 


850 


869 


889 


908 


928 


947 


966 


986 


*C05 


.3 5-1 


5.7 


224 


35 025 


044 


063 


083 


102 


121 


141 


160 


179 


199 


.4 7.8 


7.6 


225 


218 


237 


257 


276 


295 


314 


334 


353 


372 


391 


.5 9-7 


9.5 


226 


411 


430 


449 


468 


487 


507 


526 


545 


564 


583 


.611.'/ 


' 11.4 


227 


602 


621 


641 


660 


679 


698 


717 


736 


755 


774 


.7 13.6 


13.3 


228 


791 


812 


831 


850 


869 


888 


907 


926 


945 


964 


.8 15. ( 


)15.2 


229 


983 


*002 


*021 


*040 


*059 


*078 


*097 


*116 


*135 


*154 


.9 17.£ 


)17.1 


230 


36 173 


191 


210 


229 


248 


267 


286 


305 


323 


342 


-fl fi -flQ 
























lo_ lo 


231 


361 


380 


399 


417 


436 


455 


474 


492 


511 


530 


.1 l.J 


1 1.8 


232 


549 


567 


586 


605 


623 


642 


661 


679 


698 


717 


.2 3.^ 


f 3.6 


233 


735 


754 


773 


791 


810 


828 


847 


866 


884 


90S 


.3 5. J 


) 5.4 


234 


921 


940 


958 


977 


996 


*014 


*032 


*051 


*070 


*088 


.4 7-< 


t 7-2 


235 


37 107 


125 


143 


162 


180 


199 


217 


236 


254 


273 


.5 9.S 


I 9.0 


236 


291 


309 


328 


346 


364 


383 


401 


420 


438 


456 


.6 11.] 


10.8 


237 


475 


493 


511 


530 


548 


566 


58<] 


603 


621 


639 


.7 12. f 


12.6 


238 


657 


676 


694 


712 


730 


749 


767 


785 


803 


821 


.8 14. f 


5 14.4 


239 


840 
38 021 


858 
039 


876 
057 


894 
075 


912 
093 


930 
111 


948 
129 


967 


985 
165 


*003 
183 


.9 16. f 


)16.2 


240 


147 


























17 17 


241 


20T 


219 


237 


255 


273 


291 


309 


327 


345 


363 


.1 1-7 


" 1.7 


242 


381 


399 


417 


435 


453 


471 


489 


507 


525 


543 


.2 3.£ 


34 


243 


560 


578 


596 


614 


632 


650 


667 


685 


703 


721 


.3 5.2 


5.1 


244 


739 


757 


774 


792 


810 


828 


845 


8C3 


881 


899 


.4 7.C 


68 


245 


916 


934 


952 


970 


987 


*005 


*023 


*040 


*058 


*076 


.5 8.7 


8.5 


246 


39 093 


111 


129 


146 


164 


181 


199 


217 


234 


252 


.6 10. E 


10.2 


247 


269 


287 


305 


322 


340 


357 


375 


392 


410 


427 


.712.2 


11.9 


248 


445 


^f)2 


480 


497 


515 


532 


550 


567 


585 


602 


• 8 14. C 


13-6 


249 


620 


637 


655 


672 


689 


707 


724 


742 


759 


776 


.9 15.7 


15.3 


250 


794 


811 


828 
2 


846 
3 


863 
4 


881 
5 


898 
6 


915 

7 


933 


950 




N. 





1 


8 


9 


P. ] 


P. 



584 









TABLE V 


■— LOGARITHMS OF NUMBERS. 






N. 





1 


2 

828 
*002 


3 

846 

*019 


4 

863 

*036 


5 

881 

*054 


6 

898 

*071 


7 

915 


8 

933 
*105 


9 

950 
*123 


P.P. 


350 


39 794 
967 


811 
984 




251 


*088 




252 


40 140 


157 


174 


191 


209 


226 


243 


260 


277 


295 


17 17 


253 


312 


329 


346 


363 


380 


398 


415 


432 


449 


466 


.1 


1.7 


1.7 


254 


483 


500 


517 


534 


551 


569 


586 


603 


620 


637 


.2 


3.5 


3.4 


255 


654 


671 


688 


705 


722 


739 


756 


773 


790 


807 


.3 


5.2 


5.1 


256 


824 


841 


858 


875 


892 


908 


925 


942 


,959 


976 


.4 


7.0 


6.8 


257 


993 


*010 


*027 


*044 


*061 


*077 


-*^094 


*111 


*128 


*145 


.5 


8.7 


8.5 


258 


41 162 


179 


195 


212 


229 


246 


263 


279 


296 


313 


.6 


10.5 


10.2 


259 


330 


346 


363 


380 


397 


413 


430 


447 


464 


480 


•7 
.8 


12.2 
14.0 


11.9 
13-6 






















360 


497 


514 


530 


547 


564 


581 597 


614 


631 


647 


.9 


15.7 


15.3 


261 


664 


680 


697 


714 


730 


747 


764 


780 


797 


813 




262 


830 


846 


863 


880 


896 


913 


929 


946 


962 


979 




263 


995 


=^012 


*028 


*045 


*061 


*078 *094 


^111 


*127 


*144 




264 


42 160 


177 


193 


209 


226 


2421 259 


275 


292 


308 




265 


324 


341 


357 


373 


390 


4061 423 


439 


455 


472 


16 16 


266 


488 


504 


521 


537 


553 


569 586 


602 


618 


635 


.1 


1.6 


1.6 


267 


651 


667 


683 


700 


716 


732 i 748 


765 


781 


797 


.2 


3.3 


3.2 


268 


813 


829 


846 


862 


878 


894! 910 


927 


943 


959 


.3 


4.9 


48 


269 


975 


991 


*007 


*023 


*040 


*056*072 


*088 


=*=104 


*120 


.4 
.5 


6.6 
8.2 


6.4 
8.0 
























270 


43 136 


152 


168 


184 


200 


216 


233 


249 


265 


281 


.6 
.7 


9.9 
11.5 


9.6 
11.2 
























271 


297 


813 


329 


345 


361 


377 


393 


409 


425 


441 


.8 


13.2 


12.8 


272 


457 


473 


489 


505 


520 


536 


552 


568 


584 


600 


.9 


14.8 


14.4 


278 


616 


632 


648 


664 


680 


695 


711 


727 


743 


759 




274 


775 


791 


806 


822 


838 


854 


870 


886 


901 


917 




275 


933 


949 


965 


980 


996 


*012 


*028 


*043 


=^=059 


*075 




276 


44 091 


106 


122 


138 


154 


169 


185 


201 


216 


232 




277 


248 


263 


279 


295 


310 


326 


342 


357 


373 


389 




278 


404 


420 


435 


451 


467 


482 


498 


513 


529 


545 


15 15 


279 


560 


576 


591 


607 


622 


638 


653 


669 


685 


700 


.1 

.2 


1.5 
3.1 


1.5 
























30 


280 


716 


731 


747 


762 


778 


793 


809 


824 


839 


855 


.3 

.4 


4.6 
6.2 


4.5 
6.0 


281 


870 


886 


901 


917 


932 


948 


963 


978 


994 


*009 


.5 


7.7 


7.5 


282 


45 025 


040 


055 


071 


086 


102 


117 


132 


148 


163 


.6 


9.3 


9.0 


283 


178 


194 


209 


224 


240 


255 


270 


286 


301 


316 


• 7 


10.8 


10.5 


284 


332 


347 


362 


377 


393 


408 


423 


438 


454 


469 


.8 


12.4 


12.0 


285 


484 


499 


515 


530 


545 


560 


576 


591 


606 


621 


.9 


13.9 


13.5 


286 


636 


652 


667 


682 


697 


712; 727 


743 


758 


773 




287 


788 


803 


818 


833 


848 


8641 879 


894 


909 


924 




288 


939 


954 


969 


984 


999 


*014 *029 


*044 


*059 


*075 




289 


46 090 


105 


120 


135 


150 


165 


180 


195 


210 


225 




290 


240 


255 


269 


284 


299 


314 


329 


344 


359 


374 


•1| 


}\ J^- 
























1-4 


1.4 


291 


389 


404 


419 


434 


449 


464 


479 


493 


508 


523 


.2 


2.9 


2.8 


292 


538 


553 


568 


583 


597 


612 


627 


642 


657 


672 


.3 


4.3 


4.2 


298 


687 


701 


716 


731 


746 


761 


775 


790 


805 


820 


.4 


5.8 


5.6 


294 


834 


849 


864 


879 


894 


908 


923 


938 


952 


967 


.5 


7.2 


7.0 


295 


982 


997 


*011 


*026 


*041 


*055 *070 


*085 


*100 *114 


.6 


8.7 


8.4 


296 


47 129 


144 


158 


173 


188 


202 217 


232 


246 261 


.7 


10. T 


9.8 


297 


275 


290 


305 


319 


334 


348 363 


378 


392 


407 


..8 


11.6 


11.2 


298 


421 


436 


451 


465 


480 


494 509 


523 


538 


552 


.9 


13.0 


12.6 


299 


567 


581 


596 


610 


625 


639 


654 


668 


683 


697 




300 


712 


725 


741 


755 


770 


784 


799 


813 


828 


842 




N. 





1 


2 


^ 


4 


5 


6 


7 


8 


9 




P. P 





585 









TABLE v.— LOGARITHMS OF NUMBERS. 






N. 





1 

726 
871 


2 

741 
885 


3 

755 


4 

770 
914 


5 

784 
928 


6 

799 
943 


7 

813 
957 


8 

828 
972 


9 

842 
986 


P. P. 


300 


47 712 

856 




301 


900 




302 


48 000 


015 


029 


044 


058 


072 


087 


101 


115 


130 




303 


144 


158 


173 


187 


201 


216 


230 


244 


259 


273 




304 


287 


301 


316 


330 


344 


358 


373 


387 


401 


415 




305 


430 


444 


458 


472 


487 


501 


515 


529 


548 


558 




306 
307 
308 


572 
714 
855 


586 
728 
869 


600 
742 
888 


614 
756 
897 


629 
770 
911 


643 
784 
925 


657 
798 
939 


671 
812 
958 


685 
827 
967 


699 
841 
982 


• 1 
.2 
.3 
.4 
.5 
.6 
.7 
-8 
.9 


2.9 
4.3 
5.8 
7.2 

il.l 

11.6 

1 Q A 


14 

1.4 
2.8 

4.2 


309 


996 


*010 


*024 


*038 


*052 


*066 


*080 


*094 


*108 


*122 


310 


49 136 


150 


164 


178 


192 


206 


220 


234 


248 


262 


5.6 
70 


311 

312 


276 
415 


290 
429 


304 
443 


318 
457 


332 
471 


346 
485 


359 
499 


373 
513 


387 

526 


401 
540 


8.4 

9.8 

11.2 

1 n A 


313 


554 


568 


582 


596 


610 


624 


637 


651 


665 


679 


314 


693 


707 


720 


734 


748 


762 


776 


789 


803 


817 




315 


831 


845 


858 


872 


886 


900 


913 


927 


941 


955 




316 


968 


982 


996 


*010 


*023 


*037 


=*^051 


*065 


•'^078 


*092 




317 


50 106 


119 


133 


147 


160 


174 


188 


201 


215 


229 




318 


242 


256 


270 


283 


297 


311 


324 


338 


352 


365 




319 


379 


392 


406 


420 


438 


447 


460 


474 


488 


501 




330 


515 


528 


542 


555 


569 


583 


596 


610 


623 


637 




321 


650 


664 


677 


691 


704 


718 


73l 


745 


758 


772 


.1 
.2 
.3 
.4 
.5 
.6 
•7 
.8 
.9 


u 
11 

10.8 
12. T 


13 

1-3 
2.6 
3.9 
5-2 
6.5 
78 
9.1 
10.4 
11.7 


322 


785 


799 


812 


826 


839 


853 


866 


880 


898 


907 


323 


920 


933 


947 


960 


974 


987 


*001 


*014 


*027 


*041 


324 
325 
326 


51 054 
188 

322 


068 
201 
335 


081 

215 
348 


094 
228 
361 


108 
242 
375 


121 
255 
388 


135 
268 
401 


148 
282 

415 


161 
295 
428 


175 
308 
441 


327 
328 


455 
587 


468 
600 


481 
614 


494 
627 


508 
640 


521 
653 


534 
667 


547 
680 


561 
693 


574 
706 


329 


719 


733 


746 


759 


772 


785 


798 


812 


825 


838 


330 


851 
983 


864 
996 


877 
*009 


891 
*022 


904 
*035 


917 
*048 


930 
*06l 


943 


956 


969 




331 


*074 


*087 


*100 




332 


52 114 


127 


140 


153 


166 


179 


192 


205 


218 


23 




333 


244 


257 


270 


283 


296 


309 


322 


335 


348 


36 




334 


374 


387 


400 


413 


426 


439 


452 


465 


478 


49 




335 


504 


517 


530 


543 


556 


569 


582 


595 


608 


621 




336 


634 


647 


660 


672 


685 


698 


711 


724 


737 


750 


■■» ^<« 


387 


763 


776 


789 


801 


814 


827 


840 


853 


866 


879 


.1 
.2 
.3 


1.2 
2.5 
3.7 


±a 

1.2 
2.4 
3.6 


338 


891 


904 


917 


930 


943 


956 


968 


981 


994 


*007 


339 


53 020 


033 


045 


058 


071 


084 


097 


109 


122 


135 


340 


148 

275 
402 


160 

288 

415 


173 

301 
428 


186 

313 
440 


199 

326 
458 


211 

339 
466 


224 


237 


250 


262 


.4 

• 5 
-ft 

• 7 
.8 
.9 


5.0 
6.2 
7.5 
8.7 
10.0 
11.2 


4.8 
6.0 


341 
342 


352 
478 


364 
49l 


377 
504 


390 
516 


7.2 

8.4 

9.6 

10.8 


343 


529 


542 


554 


567 


580 


592 


605 


618 


630 


643 


344 


656 


668 


681 


693 


706 


719 


731 


744 


756 


769 


345 


782 


794 


807 


819 


832 


845 


857 


870 


882 


895 




346 


907 


920 


932 


945 


958 


970 


983 


995 


*008 


*020 




347 


54 033 


045 


058 


070 


083 


095 


108 


120 


133 


145 




348 


158 


170 


183 


195 


208 


220 


232 


245 


257 


270 




349 


282 


295 


307 


320 


332 


344 


357 


369 


382 


394 




350 


407 


419 


431 
2 


444 
3 


456 
4 


469 
5 


481 


493 

7 


506 
8 


518 
9 




N. 





1 


6 


P. P. 



586 









lABI 


.E V 


.—LOGARITHMS OF NUMBERS. 




N. 





1 
419 


2 

431 


3 

444 


4 

456 


5 

469 


6 

481 


7 
493 


8 
506 


9 

518 


P 


.P. 


350 


54 407 






























1^ 
























351 


53D 


543 


555 


568 


580 


592 


605 


617 


629 


642 


.1 


1.2 


352 


654 


666 


679 


691 


703 


716 


728 


740 


7531 765 


.2 


2.5 


353 


777 


790 


802 


814 


826 


839 


851 


863 


876| 888 


.3 


3.7 


354 


900 


912 


925 


937 


949 


961 


974 


986 


998*010 


.4 


5.0 


355 


55 023 


035 


047 


059 


071 


084 


096 


108 


120! 133 


.5 


6.2 


356 


145 


157 


169 


181 


194 


206 


218 


230 


242 


254 


.6 


7.5 


357 


267 


279 


291 


303 


315 


327 


340 


352 


364 


376 


.7 


8.7 


358 


388 


400 


412 


424 


437 


449 


461 


473 


485 


497 


.8 


10. 


359 


509 


521 


533 


545 


558 


570 


582 


594 


606 


618 


.9 


11.2 


360 


630 


642 


654 


666 


678 


690 


702 


714 


726 


738 






















— ■ — 










13 


361 


750 


762 


775 


787 


799 


811 


823 


835 


847 


859 


.1 


1.2 


362 


871 


883 


895 


907 


919 


931 


943 


955 


966 


978 


.2 


2.4 


363 


990 


*002 


*014 


*026 


*038 


*050 


*062 


*074 


*086,*098 


.3 


3.6 


364 


56 110 


122 


134 


146 


158 


170 


181 


193 


205 


217 


.4 


4.8 


365 


229 


241 


253 


265 


277 


288 


300 


312 


324 


336 


.5 


60 


366 


348 


360 


372 


383 


395 


407 


419 


431 


443 


455 


.6 


7.2 


367 


466 


478 


490 


502 


514 


525 


537 


549 


561 


573 


.7 


8.4 


368 


585 


596 


608 


620 


632 


643 


655 


667 


679 


691 


.8 


9.6 


369 


702 


714 


726 


738 


749 


761 


773 


785 


796 


808 


.8 


10.8 


370 


820 


832 


843 


855 


867 


879 


890 


902 


914 


925 




^1 


371 


937 


949 


961 


972 


984 


996 


*007 


*019 


*031 


*042 


.1 


372 


57 054 


066 


077 


089 


101 


112 


124 


136 


147 


159 


.2 


2.3 


373 


171 


182 


194 


206 


217 


229 


240 


252 


264 


275 


.3 


3.4 


374 


287 


299 


310 


322 


333 


345 


357 


368 


380 


391 


.4 


4.6 


375 


403 


414 


426 


438 


449 


461 


472 


484 


495 


507 


.5 


5.7 


376 


519 


530 


542 


553 


565 


576 


588 


599 


611 


622 


.6 


6-9 


377 


634 


645 


657 


668 


680 


691 


703 


714 


726 


737 


.7 


8.0 


378 


749 


760 


772 


783 


795 


806 


818 


829 


841 


852 


.8 


9.2 


379 


864 


875 


887 


898 


909 


921 


932 


944 


955 


967 


.9 


10.3 


380 


978 


990 


*001 


*012 


*024 


*035 


*047 


*058 


*069 


*081 


.1 


11 

l.± 


381 


58 092 


104 


115 


126 


138 


149 


161 


172 


183 


195 


382 


206 


217 


229 


240 


252 


263 


274 


286 


297 


308 


.2 


2.2 


383 


320 


331 


342 


354 


365 


376 


388 


399 


410 


422 


.3 


3.3 


384 


433 


444 


455 


467 


478 


489 


501 


512 


523 


535 


.4 


4.4 


385 


546 


557 


568 


580 


591 


602 


613 


625 


636 


647 


.5 


5.5 


386 


658 


670 


681 


692 


703 


715 


726 


737 


748 


760 


.6 


6.6 


387 


771 


782 


793 


804 


816 


827 


838 


849 


861 


872 


.7 


7.7 


388 


883 


894 


905 


916 


928 


939 


950 


961 


972 


984 


.8 


88 


389 


995 
59 106 


*006 


*017 
128 


*028 
140 


*039 
]'5] 


*050 
162 


*062 
173 


*073 
184 


*084 
195 


*095 
206 


.9 


9.9 


390 


117 





























1^. 


391 


217 


229 


240 


251 


262 


273 


284 


295 


306 


317 


.1 


l.Q 


392 


328 


339 


85 


362 


373 


384 


395 


40fi 


417 


428 


.2 


n 


393 


439 


450 


461 


472 


483 


494 


505 


516 


527 


538 


.3 


394 


549 


560 


571 


582 


593 


604 


615 


626 


637 


648 


.4 


4.2 


395 


659 


670 


681 


692 


703 


714 


725 


736 


747 


758 


.5 


5.2 


396 


769 


780 


791 


802 
91? 


813 


824 


835 


846 


857 


868 


.6 


6.3 


S97 


879 


890 


901 


923 


933 


944 


955 


966 


977 


.7 


7-5 


398 


988 


999 


*010 


*021 


*032 


*043 


*053 


*064 


*075 


*086 


.8 


8.4 


399 


60 097 


108 


119 


130 


141 


151 


162 


173 


184 


195 


.9 


9.3 


400 


206 


217 


227 


238 


249 


260 


271 


282 


293 


303 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P 


.P. 



587 









TABLE v.— LOGARITHMS 


OF NUMBERS. 








N. 





1 


2 


3 


4 


5 


6 


7 

282 
390 


8 

293 
401 


9 

303 
412 


P. 


P. 




400 


60 206 
3"l5 


217 


227 


238 


249 
357 


260 
368 


271 








401 


325 


336 


347 


379 




402 


422 


433 


444 


455 


466 


476 


487 


498 


509 


519 








403 


530 


541 


552 


563 


573 


584 


595 


606 


616 


627 




11 




404 


638 


649 


659 


670 


681 


692 


702 


713 


724 


735 


•1 


1.1 




405 


745 


756 


767 


777 


788 


799 


810 


820 


a3i 


842 


.2 


2 


2 




406 


852 


863 


874 


884 


895 


906 


916 


927 


938 


949 


.3 


3 


3 




407 


959 


970 


981 


991 


*002 


*013 


*023i*034| 


*044 


*055 


.4 


4 


4 




408 


61 066 


076 


087 


098 


108 


119 


130 


140 


151 


161 


.5 


5 


5 


1 


409 


172 


183 


193 


204 


215 


225 


236 


246 


257 


268 


.6 

' .7 

.8 

.9 


6 
7 
8 


6 
7 
8 




410 


278 


289 


299 


310 


320 


331 


342 


352 


363 


373 




411 


384 


394 


405 


416 


426 


437 


447 


458 


468 


479 


9 •» 




412 


489 


500 


511 


521 


532 


542 


553 


563 


574 


584 








413 


595 


605 


616 


626 


637 


647 


658 


668 


679 


689 








414 


700 


710 


721 


731 


742 


752 


763 


773 


784 


794 








415 


805 


815 


825 


836 


846 


857 


867 


878 


888 


899 




10 




416 


109 
62 al3 


920 


930 


940 


951 


961 


972 


982 


993 


*003 


.1 


1.0 




417 


024 


034 


045 


055 


065 


076 


086 


097 


107 


.2 


2.1 




418 


117 


128 


138 


149 


159 


169 


180 


190 


200 


211 


.3 


3.1 




419 


22l 


232 


242 


252 


263 


273 


283 


294 


304 


314 


.4 
.5 
.6 
.7 
.8 


4.2 
5.2 
6.3 
7.3 
8.4 


i 


430 


325 


335 


345 


356 


366 


376 


387 


397 


407 


418 




421 


428 


438 


449 


459 


469 


480 


490 


500 


510 


521 




422 


531 


541 


552 


562 


572 


582 


593 


603 


613 


624 


.9 


L 9.i 




423 


634 


644 


654 


665 


675 


685 


695 


706 


716 


726 








424 


736 


747 


757 


767 


777 


788 


798 


808 


818 


828 








425 


839 


849 


859 


869 


879 


890 


900 


910 


920 


931 








426 


941 


95.1 


961 


971 


981 


992 


*002 


*012 


*022 


*032 








427 


63 043 


053 


063 


073 


083 


093 


104 


114 


124 


134 




10 




428 


144 


154 


164 


175 


185 


195 


205 


215 


225 


235 


.1 


1.0 




429 


245 


256 


266 


276 


286 


296 


306 


316 


326 


336 


.2 
.3 
.4 
.5 
.6 


2-0 
3.0 
4.0 
5.0 
6.0 




430 


347 


357 


367 


377 


387 


397 


407 


417 


427 


437 




431 


447 


458 


468 


478 


488 


498 


508 


518 


528 


538 




432 


548 


558 


568 


578 


588 


598 


608 


618 


628 


639 


.7 


7.0 




433 


649 


659 


669 


679 


689 


699 


709 


719 


729 


739 


.8 


80 




434 


749 


759 


769 


779 


789 


799 


809 


819 


829 


839 


.9 


9.0 




435 


849 


859 


869 


879 


889 


899 


909 


919 


928 


938 








436 


948 


958 


968 


978 


988 


998 


*008 


*018 


*028 


*038 








437 


64 048 


058 


068 


078 


088 


098 


107 


117 


127 


137 








438 


147 


157 


167 


177 


187 


197 


207 


217 


226 


236 






, 


439 


246 


256 


266 


276 


286 


296 


306 


315 


325 


335 


.1 


0^ 




























440 


345 


355 


365 


375 


384 


394 


404 


414 


424 


434 


.2 
.3 


2.8 


4 


























441 


444 


453 


463 


473 


483 


493 


503 


512 


522 


532 


.4 


3.8 




442 


542 


552 


562 


571 


581 


591 


601 


611 


621 


630 


.5 


4.7 




443 


640 


650 


660 


670 


679 


689 


699 


709 


718 


72J 


• 6 


5.7 




444 


738 


748 


758 


767 


777 


787 


797 


806 


816 


826 


.7 


6.6 




445 


836 


846 


855 


865 


875 


885 


894 


904 


914 


92J 


.8 


7.6 




446 


933 


943 


953 


962 


972 


982 


992 


*001 


*011 


*02] 


.9 


8.5 




447 


65 031 


04C 


05C 


06C 


069 


079 


089 


098 


108 


lU 








448 


128 


137 


147 


157 


166 


176 


186 


195 


205 


215 








449 


224 


234 


244 


253 


263 


273 


282 


292 


302 


311 






i 


450 


821 


331 


340 


350 


36C 


369 


379 
6 


389 

7 


398 
8 


408 
9 








N. 





1 


2 


3 


4 


P. 


P. 





688 







TABLE V 


—LOGARITHMS OF NUMBERS. 






N. 





1 


2 


3 

350 


4 

360 


5 

369 


6 

379 


7 
389 


8 


9 


P. P. 


450 


65 321 


331 


340 


398 


408 




451 


417 


427 


437 


446 


456 


466 


475 


485 


494 


504 




4f)?. 


514 


523 


533 


542 


552 


562 


571 


581 


590 


600 


10 


4f)3 


610 


619 


629 


638 


648 


657 


667 


677 


686 


696 


.1 


1.0 


4f)4 


705 


715 


724 


734 


744 


753 


763 


772 


782 


791 


.2 


2.0 


4f)5 


801 


810 


820 


830 


839 


849 


858 


868 


877 


887 


.3 


30 


456 


896 


906 


915 


925 


.934 


944 


953 


963 


972 


982 


.4 


4.0 


457 


991 


*001 


*010 


*020 


*029 


*039 


=*=048 


*058 


*067 


*077 


.5 


5.0 


458 


66 086 


096 


105 


115 


124 


134 


143 


153 


162 


172 


.6 


6.0 


459 


181 
276 


190 
285 


200 
294 


209 
304 


219 


228 
323 


238 
332 


247 


257 
351 


266 
360 


.7 
.8 
.9 


7.0 


460 


313 


342 


80 
9.0 


461 


370 


379 


389 


398 


408 


417 


426 


436 


445 


455 




462 


464 


473 


483 


492 


502 


511 


520 


530 


539 


548 




463 


558 


567 


577 


586 


595 


605 


614 


623 


633 


642 




464 


652 


661 


670 


680 


689 


698 


708 


717 


726 


736 




465 


745 


754 


764 


778 


782 


792 


801 


810 


820 


829 


9 


466 


838 


848 


857 


866 


876 


885 


894 


904 


913 


922 


.1 


0.9 


467 


93l 


941 


950 


959 


969 


978 


987 


996 


*006 


*015 


• 2 


1 


• 9 


468 


67 024 


034 


043 


052 


061 


071 


080 


089 


099 


108 


.3 


2 


• 8 


469 


117 


126 


136 


145 


154 


163 


173 


182 


191 


200 


.4 
• 5 
.6 
.7 
.8 


3 
4 
5 
6 
7 


8 

.7 

l 

6 


470 


210 


219 


228 


237 


246 


256 


265 


274 


283 


293 


471 


302 


3ll 


320 


329 


339 


348 


357 


366 


376 


385 


472 


394 


403 


412 


422 


431 


440 


449 


458 


467 


477 


.9 


8 


5 


473 


486 


495 


504 


513 


523 


532 


541 


550 


559 


568 




474 


578 


587 


596 


605 


614 


623 


633 


642 


651 


660 




475 


669 


678 


687 


697 


706 


715 


724 


733 


742 


751 




476 


760 


770 


779 


788 


797 


806 


815 


824 


833 


842 




477 


852 


861 


870 


879 


888 


897 


906 


915 


924 


933 




478 


943 


952 


961 


970 


979 


988 


997 


*0C6 


*015 


*024 


9 


479 


68 033 
124 


042 
133 


051 
142 


060 
151 


.070 


079 
169 


088 
178 


097 
^87 


106 
196 


115 
205 


.1 
.2 
• 3 
.4 
.5 




1 

2. 
3. 
4. 


9 
8 

7 
6 
5 


480 


160 


481 


214 


223 


232 


241 


250 


259 


268 


277 


286 


295 


482 


304 


313 


322 


331 


340 


349 


358 


367 


376 


385 


.6 


5. 


4 


483 


394 


403 


412 


421 


430 


439 


448 


457 


466 


475 


• 7 


6. 


3 


484 


484 


493 


502 


511 


520 


529 


538 


547 


556 


565 


.8 


7. 


2 


485 


574 


583 


592 


601 


610 


619 


628 


637 


646 


654 


.9 


8. 


1 


486 


663 


672 


681 


690 


699 


708 


717 


726 


735 


744 




487 


753 


762 


770 


779 


788 


797 


806 


815 


824 


833 




488 


842 


851 


860 


868 


877 


886 


895 


904 


913 


922 




489 


931 


940 


948 


957 


966 


975 


984 


993 


*C<)2 


*016 




490 


69 019 
108 


028 


037 


046 


055 


064 


073 


081 


090 


099 


.1 
.2 




491 


117 


126 


134 


143 


152 


161 


170 


179 


187 


492 


196 


205 


214 


223 


232 


240 


249 


258 


267 


276 


.3 


2.5 


493 


284 


293 


302 


311 


320 


328 


337 


346 


355 


364 


.4 


34 


494 


372 


381 


390 


399 


408 


416 


425 


434 


443 


451 


.5 


4.2 


495 


460 


469 


478 


487 


495 


504 


513 


522 


530 


539 


.6 


5.1 


496 


548 


557 


565 


574 


583 


592 


600 


609 


618 


627 


.7 


5.9 


497 


635 


644 


653 


662 


670 


679 


688 


697 


705 


714 


.8 


68 


498 


723 


731 


740 


749 


758 


766 


775 


784 


792 


801 


• 9 


7.6 


499 


810 


819 


827 


836 


845 


853 


862 
949 

6 


871 
958 

7 


879 
966 

8 


888 
975 

9 




500 


897 


905 

1 


914 
2 


923 
3 


93l 
4 


940 
5 




N. 





P. P. 



589 







TABLE V 


—LOGARITHMS OF NUMBERS. 






V. 





1 

905 
992 


2 

914 


3 

923 


4 

93] 


5 


6 

949 
*036 


7 
958 


8 

966 
*053 


9 


P. P. 


500 


69 897 
984 


940 
*027 


975 
*06T 






501 


*001 


*010 


*018 


*044 


502 


70 070 


079 


087 


096 


105 


113 


122 


131 


139 


148 


9 




503 


157 


165 


174 


182 


191 


200 


208 


217 


226 


234 


.1 


0.9 




504 


243 


251 


260 


269 


277 


286 


294 


303 


312 


320 


.2 


1.8 




505 


329 


337 


346 


355 


363 


372 


380 


389 


398 


406 


.3 


2.7 




506 


415 


423 


432 


441 


449 


458 


466 


475 


483 


492 


• 4 


3.6 




507 


501 


509 


518 


526 


535 


543 


552 


560 


569 


578 


• 5 


4-5 




508 


586 


595 


603 


612 


620 


629 


637 


646 


654 


663 


.6 


5.4 




509 


672 


680 


689 


697 


706 


714 


723 


731 


740 


748 


.7 
.8 
.9 


6.3 
7-2 
8.1 




610 


757 


765 


774 


782 


791 


799 


808 


816 


825 


833 




511 


842 


850 


859 


867 


876 


884 


893 


901 


910 


918 






512 


927 


935 


944 


952 


961 


969 


978 


986 


995 


*003 




i 


513 


71 Oil 


020 


028 


037 


045 


054 


062 


071 


079 


088 




514 


096 


105 


113 


121 


130 


138 


147 


155 


164 


172 




J 


515 


180 


189 


197 


206 


214 


223 


231 


239 


248 


256 


8 


I 

i 


516 


265 


273 


282 


290 


298 


307 


315 


324 


332 


340 


.1 


0.8 


517 


349 


357 


366 


374 


382 


391 


399 


408 


416 


424 


.2 


1.7 




518 


433 


441 


449 


458 


466 


475 


483 


491 


500 


508 


.3 


2.5 




519 


516 


525 


533 


542 


550 


558 


567 


575 


583 


592 


.4 
.5 


3.4 

4.2 




























530 


600 


608 


617 


625 


633 


642 


650 


659 


667 


675 


.6 
.7 
.8 


5.1 
5.9 
6.8 




521 


684 


692 


700 


709 


717 


725 


734 


742 


750 


758 




522 


767 


775 


783 


792 


800 


808 


817 


825 


833 


842 


.9 


7.6 




523 


850 


858 


867 


875 


883 


891 


900 


908 


916 


925 




' 


524 


933 


941 


949 


958 


966 


974 


983 


991 


999 


*007 






525 


72 016 


024 


032 


040 


049 


057 


065 


074 


082 


090 






526 


098 


107 


115 


123 


131 


140 


148 


156 


164 


173 






527 


181 


189 


197 


206 


214 


222 


230 


238 


247 


255 






528 


263 


271 


280 


288 


296 


304 


312 


321 


329 


337 


8 




529 


345 


354 


362 


370 


378 


386 


395 


403 


411 


419 


.1 
.2 
.3 


0.8 
1.6 
2.4 


t 


530 


427 


436 


444 


452 


460 


468 


476 


485 


493 


501 


^ 

























.4 


3.2 


























531 


509 


517 


526 


534 


542 


550 


558 


566 


575 


583 


.5 


4.0 




582 


591 


599 


607 


615 


624 


632 


640 


648 


656 


664 


.6 


4.8 




533 


672 


681 


689 


697 


705 


713 


721 


729 


738 


746 


• 7 


5-6 




534 


754 


762 


770 


778 


786 


795 


803 


811 


819 


827 


.8 


6.4 




535 


835 


843 


851 


859 


868 


876 


884 


892 


900 


908 


.9 


7.2 




536 


916 


92^ 


932 


941 


949 


957 


965 


973 


981 


989 






537 


997 


*005 


*013 


*021 


*030 


*038 


*046 


*054 


*062 


*070 






538 


73 078 


086 


094 


102 


110 


118 


126 


134 


143 


151 






539 


159 


167 


175 


183 


191 


199 


207 


215 


223 


231 






540 


239 


247 


255 


263 


27l 


279 


287 


295 


303 


311 


.1 


o'. 




























541 


319 


328 


336 


344 


352 


360 


368 


376 


384 


392 


.2 


1.5 




542 


400 


408 


416 


424 


432 


440 


448 


456 


464 


472 


.3 


2.2 




543 


480 


488 


496 


504 


512 


520 


528 


536 


544 


552 


.4 


3.0 




544 


560 


568 


576 


584 


592 


600 


608 


615 


623 


631 


.5 


3.7 




545 


639 


647 


655 


663 


671 


679 


687 


695 


703 


711 


.6 


4.5 




546 


719 


727 


735 


743 


751 


759 


767 


775 


783 


791 


.7 


5.2 




547 


798 


806 


814 


822 


830 


838 


846 


854 


862 


870 


.8 


6 




548 


878 


886 


894 


902 


909 


917 


925 


933 


941 


949 


9 


6.7 




549 


957 


965 


973 


981 


989 


997 


*004 


*012 


*020 


^•'^028 






550 


74 036 


044 


052 


060 


068 


075 


083 


091 

7 


099 
8 


107 
9 




N. 





1 


2 


3 


4 


5 


6 


P. P, 



590 









TABLE v.— LOGARITHMS OF NUMBERS. 








N. 





1 


2 


3 


4 

068 


5 

075 


6 

083 


7 
09l 


8 

099 


9 


P. P. 


550 


74 036 


044 


052 


060 


107 






551 


115 


123 


131 


139 


146 


154 


162 


170 


178 


186 




552 


194 


202 


209 


217 


225 


233 


241 


249 


257 


264 






553 


272 


280 


288 


296 


304 


312 


319 


327 


335 


343 






554 


351 


359 


366 


374 


382 


390 


398 


406 


413 


421 






555 


429 


437 


445 


453 


460 


468 


476 


484 


492 


499 






556 


507 


515 


523 


531 


538 


546 


554 


562 


570 


577 






557 
558 


585 
663 


593 
671 


601 
679 


609 
687 


616 
694 


624 
702 


632 
710 


640 
718 


648 

725 


655 
733 


.1 
.2 
.3 


0.8 
1.6 
2.4 




559 


741 


749 


756 


764 


772 


780 


788 


795 


803 


811 




560 


819 


826 


834 


842 


850 


857 


865 


873 


881 


888 


• 4 
.5 


3.2 
40 




561 
562 


896 
973 


904 
98l 


912 
989 


919 
997 


927 
*004 


935 
*012 


942 
*020 


950 
*027 


958 
*035 


966 
*043 


.6 

• 7 
.8 

• 9 


48 
5.6 
64 
7.2 




563 


75 051 


058 


066 


074 


081 


089 


097 


105 


112 


120 




564 


128 


135 


143 


151 


15? 


166 


174 


182 


189 


197 




565 


205 


212 


220 


228 


235 


243 


251 


258 


266 


274 






566 


281 


289 


297 


304 


312 


320 


327 


335 


343 


350 






567 


358 


366 


373 


381 


389 


396 


404 


412 


419 


427 






568 


435 


442 


450 


458 


465 


473 


480 


488 


496 


503 






569 


511 


519 


526 


534 


541 


549 


557 


564 


572 

648 

724 
800 
876 


580 

656 

732 
808 
883 






570 


587 

663 
739 
815 


595 

671 
747 
823 


602 

679 
755 
830 


610 

686 
762 
838 


618 

694 
770 
846 


625 


633 

709 
785 
861 


641 

717 
792 
868 




571 
572 
573 


701 
777 
853 


.1 
.2 
.3 
.4 
.5 
• 6 
7 
.8 
.9 



1 
2 
3 
3 
4 
5 
6 
6 


'7 

5 

?■ 
( 
7 

1 




574 


891 


899 


906 


914 


921 


929 


936 


944 


951 


959 




575 


967 


974 


982 


989 


997 


=^004 


*012 


*019 


*027 


*03^ 




576 
577 


76 042 
117 


050 
125 


057 
132 


065 
140 


072 
147 


080 
155 


087 
162 


095 
170 


102 
178 


110 
185 




578 


193 


200 


208 


215 


223 


230 


238 


245 


253 


260 




579 


268 
343 
417 


275 


283 


29d 


298 


305 


313 


320 


328 


335 




580 


350 
425 


358 

432 


365 
440 


372 
447 


380 


387 


395 


402 


410 






581 


455 


462 


470 


477 


485 




582 


492 


500 


507 


514 


522 


529 


537 


544 


552 


559 






583 


567 


574 


582 


589 


596 


604 


611 


619 


626 


634 






584 


641 


648 


656 


663 


671 


678 


686 


693 


700 


708 






585 


715 


723 


730 


738 


745 


752 


760 


767 


775 


782 






586 


790 


797 


804 


812 


819 


827 


834 


841 


849 


856 






587 


864 


871 


878 


886 


893 


901 


908 


915 


923 


930 


.1 
.2 
• 3 


0.7 
1.4 
2.1 




588 


937 


945 


952 


960 


967 


974 


982 


989 


997 


*004 




589 


77 Oil 


019 


026 


033 


041 


048 


055 


063 


070 


078 




590 


085 


092 


100 


107 


114 


122 


129 


136 


144 


151 


.4 
.5 


2.8 

3.5 




591 


158 


166 


173 


181 


188 


195 


203 


210 


217 


225 


.6 
• 7 
.8 
.9 


4.2 
4.9 
56 
6.3 




592 


232 


239 


247 


254 


261 


269 


276 


283 


291 


298 




593 
594 


30 
378 


313 
386 


320 
393 


327 
400 


335 
408 


342 
415 


349 
422 


356 
430 


364 
437 


371 

444 




595 


45; 


459 


466 


473 


481 


488 


495 


503 


510 


517 






596 


52; 


532 


539 


546 


554 


561 


568 


575 


583 


590 






597 


597 


604 


612 


619 


626 


634 


641 


648 


655 


663 






598 


670 


677 


684 


692 


699 


706 


713 


721 


728 


735 






599 


742 


750 


757 


764 


771 


779 


786 


793 


800 


808 






600 


815 


822 


829 
2 


837 
3 


844 
4 


851 


858 


866 


873 


880 




N. 





1 


5 


6 


7 


8 


9 


P. 


P. 







591 



TABLE V.—LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 

837 


4 

844 


5 

85l 


6 

858 


7 
866 


8 
873 


9 

880 


P 


. P. 


600 


77 815 
887 


822 


829 






601 


894 


902 


909 


916 


923 


931 


938 


945 


952 




602 


95{ 


967 


974 


981 


988 


995 


*003 


*010 


*017 


*024 






603 


78 03: 


039 


046 


053 


060 


067 


075 


082 


089 


096 






604 


103 


111 


118 


125 


132 


139 


147 


154 


161 


16» 






605 


17 


182 


190 


197 


204 


211 


218 


226 


233 


240 






606 


247 


254 


261 


269 


276 


283 


290 


297 


304 


311 




1.5 
2-2 


607 
608 


319 
390 


326 
397 


333 

404 


340 
412 


347 
419 


354 
426 


362 
433 


369 

440 


376 
447 


383 

454 


.1 

.2 
.3 


609 


461 


469 


476 


483 


490 


497 


504 


511 


518 


526 


610 


533 


540 


547 


554 


561 


568 


575 


583 


590 


597 


.4 
.5 


'd 


611 


604 


611 


618 


625 


632 


639 


646 


654 


661 


668 


.6 
.7 
.8 
.9 


4.5 
5.2 


612 


675 


682 


689 


696 


703 


710 


717 


725 


732 


739 


613 


746 


753 


760 


767 


774 


781 


788 


795 


802 


810 


614 


817 


824 


831 


838 


845 


852 


859 


866 


873 


880 


615 


887 


894 


901 


908 


915 


923 


930 


937 


944 


951 






616 


958 


965 


972 


979 


986 


993 


*000 


*007 


*014 


*021 






617 


79 028 


035 


042 


049 


056 


063 


070 


078 


085 


092 






618 


099 


106 


113 


120 


127 


134 


141 


148 


155 


162 






619 


169 


176 


183 


190 


197 


204 


211 


218 


225 


232 






630 


239 


246 


253 


260 


267 


274 


281 


288 


295 


302 






621 


309 


316 


323 


330 


337 


344 


351 


358 


365 


372 


.1 

.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


0^7 

1-4 
2.1 
2.8 
3.5 
4.2 
4.9 
5.6 
6.3 


622 
623 
624 


379 
449 
518 


386 

456 
525 


393 
462 
532 


400 
469 
539 


407 
476 
546 


414 
483 
553 


421 
490 
560 


428 
497 
567 


435 
504 
574 


442 
511 
581 


625 


588 


595 


602 


609 


616 


622 


629 


636 


643 


650 


626 
627 


657 
727 


664 
733 


671 
740 


678 

747 


685 

754 


692 
761 


699 
768 


706 
775 


713 
782 


720 
789 


628 


796 


803 


810 


816 


823 


830 


837 


844 


851 


858 


629 


865 


872 


879 


886 


892 


899 


906 


913 


920 


927 


630 


934 
80 003 


941 
010 


948 
016 


954 
023 


961 
030 


968 
037 


975 


982 


989 


996 






631 


044 


051 


058 


065 




632 


07l 


078 


085 


092 


099 


106 


113 


120 


126 


133 






633 


140 


147 


154 


161 


168 


174 


181 


188 


195 


202 






634 


209 


216 


222 


229 


236 


243 


250 


257 


263 


270 






635 


277 


284 


291 


298 


304 


311 


318 


325 


332 


339 






636 


345 


352 


359 


366 


373 


380 


386 


393 


400 


407 




13 
1.9 


637 
638 


414 
482 


421 
489 


427 
495 


434 
502 


441 
509 


448 
516 


455 
523 


461 

529 


468 
536 


475 
543 


.1 

• 2 

• 3 


639 


550 


557 


563 


570 


577 


584 


591 


597 


604 


611 


640 


618 
686 


625 


631 


638 


645 


652 


658 
726 


665 
733 


672 
740 


679 

746 


.4 
.5 
.6 
.7 
.8 
.9 


2.6 
3.2 


641 


692 


699 


706 


713 


719 


39 
4.5 
5.2 
5.8 


642 


753 


760 


767 


774 


780 


787 


794 


801 


807 


814 


643 
644 


821 
888 


828 
895 


834 
902 


841 
909 


848 

915 


855 

922 


861 
929 


868 
936 


875 

942 


882 
949 


645 


956 


962 


969 


976 


983 


989 


996 


*003 


*010 


*016 






646 


81 023 


030 


036 


043 


050 


057 


063 


070 


077 


083 






647 


090 


097 


104 


110 


117 


124 


130 


137 


144 


151 






648 


157 


164 


171 


177 


184 


191 


197 


204 


211 


218 






649 


224 


231 


238 


244 


251 


258 


264 


271 


278 


284 






650 


29l 


298 


304 
2 


3ll 
3 


318 
4 


324 
5 


331 
6 


338 

7 


345 
8 


35l 
9 






N. 





1 


V 


P. 



592 









lABLE V 


.—LOGARITHMS OF NUMBERS. 






N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P 


. P. 


650 


81 29l 


298 


304 


311 


318 


324 


33l 


338 


345 


351 






651 


358 


365 


371 


378 


385 


391 


398 


405 


411 


418 




652 


425 


431 


438 


444 


451 


458 


464 


471 


478 


484 






653 


491 


498 


504 


511 


518 


524 


531 


538 


544 


551 






654 


558 


564 


571 


577 


584 


591 


597 


604 


611 


617 






655 


624 


631 


637 


644 


650 


657 


664 


670 


677 


684 






656 


690 


697 


703 


710 


717 


723 


730 


736 


743 


750 




7 
0.7 
1.4 
2.1 


657 


756 


763 


770 


776 


783 


789 


796 


803 


809 


816 


.1 
.2 
.3 


658 


822 


829 


836 


842 


849 


855 


862 


869 


875 


882 


659 


888 


895 


901 


908 


915 


921 


928 


934 


941 


948 


660 


954 


961 


967 


974 


980 


987 


994 


*000 


*007 


*013 


.4 
.5 


2.8 
3.5 


661 
662 


82 020 
086 


026 
092 


033 
099 


040 
105 


046 
112 


053 
118 


059 
125 


066 
131 


072 
138 


079 
145 


.6 
.7 
.8 
.9 


4.2 
4.9 
5.6 
6.3 


663 
664 


151 
217 


158 

223 


164 
230 


171 
236 


177 
243 


184 

249 


190 
256 


197 
262 


203 
269 


210 
275 


665 


282 


288 


295 


302 


308 


315 


321 


328 


334 


341 






666 


347 


354 


360 


367 


373 


380 


386 


393 


399 


406 






667 


412 


419 


425 


432 


438 


445 


451 


458 


464 


471 






668 


477 


484 


490 


497 


503 


510 


516 


523 


529 


536 






669 


542 
607 


549 
614 


555 
620 


562 
627 


568 
633 


575 
640 


581 
646 


588 


594 
659 


601 
666 






670 


653 




671 


672 


678 


685 


691 


698 


704 


711 


717 


724 


730 


.1 

.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


0^- 


672 


737 


743 


750 


756 


763 


769 


775 


782 


788 


795 


1 
1 
2 
3 
3 
4 
5 
5 


o 
3 

9 

9 
5 

i 


673 


801 


808 


814 


821 


827 


834 


840 


846 


853 


859 


674 


866 


872 


879 


885 


892 


898 


904 


91] 


917 


924 


675 


930 


937 


943 


949 


956 


962 


969 


975 


982 


988 


676 


994 


*001 


*007 


*014 


*020 


*027 


*033 


*039 


*046 


*052 


677 


83 059 


065 


071 


078 


084 


091 


097 


103 


110 


116 


678 
679 


123 
187 


129 
193 


136 
200 


142 
206 


148 
212 


155 
2i^ 


161 
225 


168 
231 


174 
238 


180 

244 


680 


251 


257 


263 


270 


276 


283 


289 


295 


302 


308 






681 


314 


321 


327 


334 


340 


346 


353 


359 


365 


372 




682 


378 


385 


391 


397 


404 


410 


416 


423 


429 


435 






683 


442 


448 


455 


461 


467 


474 


480 


486 


493 


499 






684 


505 


512 


518 


524 


531 


537 


543 


550 


556 


562 






685 


569 


575 


581 


588 


594 


600 


607 


613 


619 


626 






686 


632 


638 


645 


651 


657 


664 


670 


676 


683 


689 




6 

0.6 
1.2 
1-8 


687 


695 


702 


708 


714 


721 


727 


733 


740 


746 


752 


.1 
.2 
.3 


688 


759 


765 


771 


778 


784 


790 


796 


803 


809 


815 


689 


822 


828 


834 


841 


847 


853 


859 


866 


872 


878 


690 


885 


891 


897 


904 


910 


916 


922 


929 


935 


941 


.4 
.5 


2.4 
3.0 


691 


948 


954 


960 


966 


973 


979 


985 


992 


998 


*004 


.6 
.7 
.8 
.9 


3.6 
4.2 
4.8 
5.4 


692 


84 010 


017 


023 


029 


035 


042 


048 


054 


061 


067 


693 

694 


073 
136 


079 
142 


086 
148 


092 
154 


098 
161 


104 
167 


111 
173 


117 
179 


123 
186 


129 
192 


695 


198 


204 


211 


217 


223 


229 


236 


242 


248 


254 






696 


261 


267 


273 


279 


286 


292 


298 


304 


311 


317 






697 


323 


329 


335 


342 


348 


354 


360 


367 


373 


379 






698 


385 


392 


398 


404 


410 


416 


423 


429 


435 


441 






699 


447 
510 


454 
516 


460 


466 


472 


479 


485 


491 


497 


503 






700 


522 


528 
3 


534 
4 


541 
5 


547 
6 


553 


559 


565 




N. 





1 


2 


7 


8 


9 


P. 


P. 





593 









TABLE v.— LOGARITHMS OF NUMBERS. 




N. 





1 
516 


2 

522 


3 

528 


4 

534 


5 

541 


6 

547 


7 
553 


8 
559 


9 

565 


P. P. 


700 


84 510 




701 
702 
703 
704 
705 
706 
707 
708 
709 


572 
633 
695 
757 
819 
880 
942 
85 003 
064 


578 
640 
701 
763 
825 
886 
948 
009 
070 


584 
646 
708 
769 
831 
893 
954 
015 
077 


590 
652 
714 
776 
837 
899 
960 
021 
083 


596 
658 
720 
782 
843 
905 
966 
028 
089 


603 
664 
726 
788 
849 
911 
972 
034 
095 


609 
671 
732 
794 
856 
917 
979 
040 
lOl 


615 
677 
739 
800 
862 
923 
985 
046 
107 


621 
683 
745 
806 
868 
929 
991 
052 
113 


62.7 
689 
751 
813 
874 
936 
997 
058 
119 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 

.1 
.2 
.3 
• 4 
.5 

1 

.8 

.9 


1.3 
1.9 


710 


126 

187 
248 
309 
370 
430 
491 
552 
612 
673 

733 

793 
853 
914 
974 
86 034 
093 
153 
213 
273 


132 

193 
254 
315 
376 
436 
497 
558 
618 
679 

739 

799 
859 
920 
980 
040 
099 
159 
219 
278 


138 

199 
260 
321 
382 
443 
503 
564 
624 
685 

745 

805 
865 
926 
986 
046 
105 
165 
225 
284 


144 


150 


156 


162 


16§ 


174 


181 


2.6 
3.2 


711 
712 
713 
714 
715 
716 
717 
718 
719 


205 
266 
327 
388 
449 
509 
570 
630 
691 


211 
272 
333 
394 
455 
515 
576 
636 
697 


217 
278 
339 
400 
461 
521 
582 
642 
703 


223 
284 
345 
406 
467 
527 
588 
648 
709 


229 
290 
351 
412 
473 
533 
594 
655 
715 


236 
297 
357 
418 
479 
540 
600 
661 
721 


242 
303 
36£ 
42i 
485 
546 
606 
667 
727 


5.2 
5.8 


730 

721 
722 
723 
724 
725 
726 
727 
728 
729 


751 

811 
872 
932 
992 
052 
111 
171 
231 
290 


757 

817 
878 
938 
998 
058 
117 
177 
237 
296 


763 

823 
884 

944 
*004 
063 
123 
183 
243 
302 


769 

829 
890 
950 
*010 
069 
129 
189 
249 
308 


775 

835 
896 
956 
*016 
075 
135 
195 
255 
314 


781 

841 

902 

962 

*022 

08:: 

141 
201 
261 
320 


787 

847 
908 
968 
*028 
087 
147 
207 
267 
326 


6 

0.6 
1.2 
1.8 
2.4 
3.0 
3.6 
4.2 
4.8 
5.4 


730 


332 


338 


344 


350 


356 


362 


368 


374 


380 


386 




731 
732 
733 
734 
735 
736 
737 
738 
739 


391 
451 
510 
569 
628 
688 
746 
805 
864 


397 
457 
516 
575 
634 
695 
752 
811 
870 


40§ 
463 
522 
581 
640 
699 
758 
817 
876 


409 
469 
528 
587 
646 
705 
764 
823 
882 


415 
475 
534 
593 
652 
711 
770 
829 
888 


421 
481 
540 
599 
658 
717 
776 
835 
894 


427 
486 
546 
605 
664 
723 
782 
841 
899 


433 
492 
552 
611 
670 
729 
788 
847 
905 


439 
498 
558 
617 
676 
735 
794 
852 
911 


445 
504 
563 
623 
682 
741 
800 
858 
917 


• 1 
.2 

• 3 
.4 
.5 
.6 
.7 
.8 
.9 


0^5 


740 


923 

982 
87 040 
099 
157 
215 
274 
332 
390 
448 

506 


929 

987 
046 
104 
163 
221 
279 
338 
396 
454 

512 


935 

993 
052 
110 
169 
227 
285 
343 
402 
460 

517 
2 


941 

999 
058 
116 
175 
233 
291 
349 
407 
465 

523 
3 


946 


952 


958 


964 


970 


976 


2.2 
2.7 


741 
742 
743 
744 
745 
746 
747 
748 
749 

760 


*005 
064 
122 
180 
239 
297 
355 
413 
471 

529 
4 


*011 
069 
128 
186 
245 
303 
361 
419 
477 

535 
5 


*017 
075 
134 
192 
250 
309 
367 
425 
483 

541 
6 


*023 
081 
140 
198 
256 
314 
372 
431 
489 

546 

7 


*028 
087 
145 
204 
262 
320 
378 
436 
494 

552 
8 


*034 
093 
151 
210 
268 
326 
384 
442 
500 

558 
9 


3.3 
3.8 

t1 


N. 





1 


P. P. 



694 









TABLE 7.— LOGARITHMS 


OF NUMBERS. 




N. 





1 


2 


3 

523 
581 


4 

529 
587 


5 

535 


6 

541 


7 


8 


9 


P. P. 


750 


87 506 
564 


512 
570 


517 
575 


546 


552 


558 




751 


593 


598 


604 


610 


616 




752 


622 


627 


633 


639 645 


650 


656 


662 


668 


673 




753 


679 


685! 691 


6971 702 


708 


714 


720 


725 


731 




754 


737 


1 743: 748 


1 754' 760 


766 


771 


777 


783 


789 




755 


794 


800: 806 


812; 817 


823 


; 829 


835 


840 


846 




756 


852 


858 863 


869! 875 


881 


1 886 


892 


898 


904 


^ 


757 


909 


915 


921 


927 


932 


938 


944 


949 


955 


961 


.1 
.2 
.3 
.4 
.5 


o 

0.6 
1.2 
1-8 


758 


967 


972 


978 


984 


990 


995*001 


*007 


*012 


*018 


759 


88 024 


030 


035 


041 


047 


053 


; 058 


064 


070 


075 


760 


081 


087 


093 


098 


104 


110 


115 


121 


127 


133 


2-4 
30 


761 


138 


! 144 


150 


155 


161 


167 


172 


178 


184 


190 


.6 
.7 
.8 
.9 


3.6 

4-2 
4.8 
5.4 


762 


195 


j 201 


207 


21§ 


218 


224 


229 


235 


241 


247 


763 


252 


258 


264 


269 


275 


281 


286 


292 


298 


303 


764 


309 


I 315 


320 


326 


332 


337 


343 


349 


355 


360 


765 


366 


: 372 


377 


383 


389 


394 


400 


406 


411 


417 




766 


423 


428 


434 


440 


445 


451 


457 


462 


468 


474 




767 


479 


485 


491 


496 


502 


508 


513 


519 


525 


530 




768 


536 


542 


547 


553 


558 


564 


570 


575 


581 


587 




769 


595 


598 


604 


609j 615 


621 


626 


632 


638 


643 




770 


649 
705 


654 
711 


660 


666 


671 
728 


677 
733 


683 
739 


688 
745 


694 


700 
756 




771 


716 


722 


750 


.1 
.2 
• 3 
.4 
.5 
.6 
.7 
.8 
.9 


0^ 

1.1 

1.6 

i;l 

3.3 
3.8 


772 


761 


767 


773 


778 


784 


790 


795 


801 


806 


812 


773 


818 


823 


829 


835 


840 


846 


851 


857 


863 


868 


774 


874 


879 


885 


891 


896 


902 


907 


913 


919 


924 


775 


930 


936 


941 


947! 952 


958 


964 


969 


975 


980 


776 


986 


992 


997 


*003 *008 


=^014 


*019;*025 


*031 


*036 


777 


89 042 


047 


053 


0591 064 


070 


075 


081 


087 


092 


778 


098 


103 


109 


114 120 


126 


131 


137 


142 


148 


779 


153 


159 


165 


170 


176 


181 


187 


193 


198 


204 


780 


209 


215 


220 


226 


231 


237 


243 


248 


254 


259 




781 


265 


270 


276 


282 


287 


293 


298 


304 


309 


315 




782 


320 


326 


332 


337 


343 


348 


354 


359 


365 


370 




783 


376 


381 


387 


393 


398 


404 


409 


415 


429 


426 




784 


431 


437 


442 


448 454 


459 


465 


470 


476 


481 




785 


487 


492 


498 


503 509 


514 


520 


525 


531 


536 




786 


542 


548 


553 


559, 564 


570 


575 


581 


586 


592 


e> 


787 


597 


603 


608 


614' 619 


625 


630 


636 


64l 


647 


.1 

.2 
.3 
.4 
.5 
.6 

-.1 

.9 


0.5 
1.0 

1.5 


788 


652 


658 


663 


669 674 


680 


685 


691 


696 


702 


789 


707 


713 


718 


724! 729 


735 


740 


746 


751 


757 


790 


762 
817 


768 
823 


773 


779 784 
834 839 


790 
845 


795 
850 


801 
856 


806 
861 


812 
867 


2.0 

2.5 


791 


828 


30 
3.5 
4.0 
4.5 


792 


872 


878 


883 


889 894 


900 


905 


911 


916 


922 


793 


927 


933 


938 


943 i 949 


954 


960 


965 


971 


976 


794 


982 


987 


993 


998*004 


*009 *015| 


*020 


*026 


*031 


795 


90 036 


042 


047! 


053 058 


064 


069 


075 


080 


086 




796 


091 


097 


102 


107 


113 


118, 


124 


129 


135 


140 




797 


146 


151 


156 


162 


167 


173' 


178 


184 


189 


195 




798 


200 


205 


211 


216 


222 


2271 


233 


238 


244 


249 




799 


254 


260 


265 


271 


276 


282 


287 


292 


298 


303 




800 


809 


314 

1 


320 
2 


325 
3 


330 
4 


336 


341 
6 


347 

7 


352 
8 


358 
9 




N. 





5 


P. P. 



595 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 

341 


7 
347 


8 

352 


9 

358 


P. P. 


800 


90 309 


314 


320 


325 


330 


336, 




801 
802 
803 
804 
805 
806 
807 
808 
809 


363 

417 

47l 

52 

579 

633 

687 

741 

795 


368 
423 
477 
531 
585 
639 
692 
746 
800 


374 
428 
482 
536 
590 
644 
698 
752 
805 


379 
433 
488 
542 
596 
649 
703 
757 
811 


385 
439 
493 
547 
601 
655 
709 
762 
816 


390 
444 
498 
552 
606 
660 
714 
768 
821 


396 
450 
504 
558 
612 
666 
719 
773 
827 


401 
455 
509 
563 
617 
671 
725 
778 
832 


406 
460 
515 
569 
622 
676 
730 
784 
838 


412 
466 
520 
574 
628 
682 
736 
789 
843 




810 


848 


854 


859 


864 


870 


875 


880 


886 


891 


896 




811 
812 
813 
814 
815 
816 
817 
818 
819 


902 
955 
91 009 
062 
116 
169 
222 
275 
328 


907 
961 
014 
068 
121 
174 
227 
280 
333 


913 
966 
019 
073 
126 
179 
233 
286 
339 


918 
971 
025 
078 
131 
185 
238 
291 
344 


923 
977 
030 
084 
137 
190 
243 
296 
349 


929 
982 
036 
089 
142 
195 
249 
302 
355 


934 
987 
041 
094 
147 
201 
254 
307 
360 


939 
993 
046 
100 
153 
206 
259 
312 
365 


945 
998 
052 
105 
158 
211 
264 
318 
371 


950 
*003 
057 
110 
163 
217 
270 
323 
376 




.1 
.2 
.3 
.4 
.5 
6 
.7 
8 
9 


0^5 

1.1 

1.6 

U 

3-3 
3.8 


830 


381 


386 


392 


397 


402 


408 


413 


418 


423 


429 




821 
822 
823 
824 
825 
826 
827 
828 
829 

830 


434 
487 
540 
592 
645 
698 
750 
803 
855 

908 

960 
92 012 
064 
116 
168 
220 
272 
324 
376 


439 
492 
545 
598 
650 
703 
756 
808 
860 

913 

965 
017 
069 
122 
174 
226 
277 
329 
381 


445 
497 
550 
603 
656 
708 
761 
813 
866 

918 


450 
503 
556 
608 
661 
714 
766 
819 
871 

923 

976 
028 
080 
132 
184 
236 
288 
340 
391 


455 
508 
561 
614 
666 
719 
771 
824 
876 

928 

981 
033 
085 
137 
189 
241 
293 
345 
397 


461 
513 
566 
619 
671 
724 
777 
829 
881 

934 

986 
038 
090 
142 
194 
246 
298 
350 
402 


466 
519 
571 
624 
677 
729 
782 
834 
887 

939 

99l 
043 
096 
148 
200 
252 
303 
355 
407 


471 
524 
577 
629 
682 
735 
787 
839 
892 

944 

996 
049 
101 
153 
205 
257 
309 
360 
412 


476 
529 
582 
635 
687 
740 
792 
845 
897 

949 

*002 
054 
106 
158 
210 
262 
314 
366 
417 


482 
534 
587 
640 
692 
745 
798 
850 
902 

955 

*007 
059 
111 
163 
215 
267 
319 
371 
423 




831 
832 
833 
834 
835 
836 
837 
838 
839 


970 
023 
075 
127 
179 
231 
283 
335 
386 




1 
2 
3 
4 
5 
6 
7 
8 
9 


5 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 


840 

841 
842 
843 
844 
845 
846 
847 
848 
849 

850 


428 

479 
531 
583 
634 
685 
737 
788 
839 
891 

942 


433 

485 
536 
588 
639 
691 
742 
793 
844 
896 

947 


438 

490 
541 
593 
644 
696 
747 
798 
850 
901 

952 


443 

495 
546 
598 
649 
701 
752 
803 
855 
906 

957 


448 

500 
552 
603 
655 
706 
757 
809 
860 
911 

962 
4 


454 

505 
557 
608 
660 
711 
762 
814 
865 
916 

967 
5 


459 

510 
562 
613 
665 
716 
768 
819 
870 
921 

972 
6 


464 

515 
567 

6ia 

670 
721 
773 
824 
875 
926 

977 

7 


469 

521 
572 
624 
675 
727 
778 
829 
880 
931 

982 
8 


474 

526 
577 
629 
680 
732 
783 
834 
885 
937 

988 




N. 





1 


2 


3 


9 


P. P. 



596 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 


3 

952 


3 

957 


4 

962 


5 

967 


6 

972 


7 
977 


8 


9 


P. P. 


850 


92 942 


947 


982 


988 




851 


993 


998 


*003 


*008 


*013 


*018 


*023 


*028 


*034 


*039 




852 


93 044 


049 


054 


059 


064 


069 


074 


079 


084 


090 




853 


095 


100 


105 


110 


115 


120 


125 


130 


135 


140 




854 


146 


151 


156 


161 


166 


171 


176 


181 


186 


191 




855 


196 


201 


207 


212 


217 


222 


227 


232 


237 


242 




856 


247 


252 


257 


262 


267 


272 


278 


283 


288 


293 


•F 


857 
858 
859 


298 
348 
399 


303 
354 
404 


308 
359 
409 


313 
364 
414 


318 
369 
419 


323 
374 
424 


328 
379 
429 


333 
384 

434 


338 
389 

439 


343 
394 
445 


.1 
.2 
.3 


1.1 

1.6 


860 


450 


455 


460 


465 


470 


475 


480 


485 


490 


495 


.4 
.5 
.6 
.7 
.8 
.9 


2-2 
2.7 


861 


500 


505 


510 


515 


520 


525 


530 


535 


540 


545 


3-3 
3-8 

4.4 
4.9 


862 


550 


556 


561 


566 


571 


576 


58: 


586 


591 


596 


863 


601 


606 


611 


616 


621 


626 


63 


636 


64 


646 


864 


651 


656 


661 


666 


671 


676 


681 


686 


691 


696 


865 


701 


706 


711 


716 


721 


726 


73 


736 


742 


747 




866 


752 


757 


762 


767 


772 


777 


782 


787 


792 


797 




867 


802 


807 


812 


817 


822 


827 


832 


837 


842 


847 




868 


852 


857 


862 


867 


872 


877 


882 


887 


892 


897 




869 


902 


907 


912 


917 


922 


927 


932 


937 


942 


947 




870 


952 


957 


962 


967 


972 


977 


982 


987 


992 


997 




871 


94 002 


007 


012 


017 


022 


026 


031 


036 


041 


046 


.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 


5 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 


872 


051 


056 


061 


066 


071 


076 


081 


086 


091 


096 


873 


lOl 


106 


111 


116 


121 


126 


131 


136 


141 


146 


874 


151 


156 


161 


166 


171 


176 


181 


186 


191 


196 


875 
876 
877 
878 
879 


201 
250 
300 
349 
399 


206 
255 
305 
354 
404 


210 
260 
310 
359 
409 


215 
265 
315 
364 
413 


220 
270 
320 
369 
418 


223 
275 
324 
374 
428 


230 
280 
329 
379 
428 


235 
285 
334 
384 
433 


240 
290 
339 
389 
438 


245 
295 
344 
394 
443 


880 


448 
497 


453 
502 


458 

507 


463 
512 


468 

517 


473 

522 


478 

527 


483 

532 


487 
537 


492 




881 


542 




882 


547 


552 


556 


561 


566 


571 


576 


581 


586 


591 




883 


596 


601 


606 


611 


615 


620 


625 


630 


635 


640 




884 


645 


650 


655 


660 


665 


670 


674 


679 


684 


689 




885 


694 


699 


704 


709 


714 


719 


724 


728 


733 


738 




886 


743 


748 


753 


758 


763 


768 


773 


777 


782 


787 


T 


887 
888 


792 
841 


797 
846 


802 
851 


807 
856 


812 
861 


817 
865 


821 
870 


826 
875 


831 
880 


836 
885 


• 1 

• 2 
3 




1 


\ 


889 


890 


895 


900 


905 


909 


914 


919 


924 


929 


934 


890 


939 


944 


949 


953 


958 


963 


968 


973 


978 


983 


.4 
.5 


1 
2 


1 


891 
892 


988 
95 036 


992 
04l 


997 
046 


*002 
051 


*007 
058 


*012 
061 


*017 
065 


*022 
070 


*026 
075 


031 
080 


.6 
.7 
.8 
.9 


2 
3 
3 
4 


1 


893 
894 


085 
134 


090 
138 


095 
143 


099 
148 


104 
153 


109 
158 


114 
163 


119 
167 


124 
172 


129 
177 


895 


182 


187 


192 


197 


201 


206 


211 


216 


221 


226 




896 


231 


235 


240 


245 


250 


255 


260 


264 


269 


274 




897 


279 


284 


289 


294 


298 


303 


308 


313 


318 


323 




898 


327 


332 


337 


342 


347 


352 


356 


361 


366 


371 




899 


376 


381 


385 


390 


395 


400 


405 


410 


414 


419 




900 


424 


429 


434 


438 


443 


448 


453 


458 


463 


467 




N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 



597 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 


2 

434 


3 

438 


4 

443 


5 

448 


6 

453 


7 
458 


8 
463 


9 


P. P. 


900 


95 424 


429 


467 




901 


472 


477 


482 


487 


492 


496 


50l 


506 


511 


516 




902 


520 


525 


530 


535 


540 


544 


549 


554 


559 


564 




903 


569 


573 


578 


583 


588 


593 


597 


602 


607 


612 




904 


617 


621 


626 


631 


636 


641 


645 


650 


655 


660 




905 


665 


669 


674 


679 


684 


689 


693 


698 


703 


708 




906 


713 


717 


722 


727 


732 


737 


741 


746 


751 


756 




907 


760 


765 


770 


775 


780 


784 


789 


794 


799 


804 




908 


808 


813 


818 


823 


827 


832 


837 


842 


847 


851 




909 


856 


861 


866 


870 


875 


880 


885 


890 


894 


899 




910 


904 
952 


909 
956 


913 
96l 


918 
966 


923 


928 
975 


933 
980 


937 
985 


942 
990 


947 
994 




911 


971 


5 


912 
913 
914 
915 


999 

96 047 

094 

142 


*004 
052 
099 
147 


*009 
056 
104 
151 


*014 
061 
109 
156 


*018 
066 
113 
161 


*023 
071 
118 
166 


*028 
075 
123 
170 


*033 
080 
128 
175 


*037 
085 
132 
180 


*042 
090 
137 
]85 


•1 
.2 

• 3 

• 4 
.5 
.6 

• 7 
.8 
.9 


1.0 
1.5 
2.0 
2.5 
30 
3.5 
40 
4.5 


916 
917 


189 
237 


194 
24l 


199 

246 


204 
251 


208 
256 


213 
260 


218 
265 


222 
270 


227 
275 


232 
279 


918 
919 


284 
33l 


289 

336 


293 
341 


298 
345 


303 
350 


308 
355 


312 
360 


317 
364 


322 
369 


327 
374 


930 


379 


383 


388 


393 


397 


402 


407 


412 


416 


421 




921 


426 


430 


435 


440 


445 


449 


454 


459 


463 


468 




922 


473 


478 


482 


487 


492 


496 


501 


506 


511 


515 




923 


520 


525 


529 


534 


539 


543 


548 


553 


558 


562 




924 


567 


572 


576 


581 


586 


590 


595 


600 


605 


609 




925 


614 


619 


623 


628 


633 


637 


642 


647 


651 


656 




926 


661 


666 


670 


675 


680 


684 


689 


694 


698 


703 




927 


708 


712 


717 


722 


726 


731 


736 


741 


745 


750 




928 


755 


759 


764 


769 


773 


778 


783 


787 


792 


797 




929 


801 


806 


811 


815 


820 


825 


829 


834 


839 


843 




930 


848 


853 


857 


862 


867 


87l 


876 


881 


885 


890 




931 
932 


895 
94l 


899 
946 


904 
95] 


909 
955 


913 
960 


918 
965 


923 
969 


927 
974 


932 
979 


937 
983 


.1 
.2 
.3 
• 4 
.5 
.6 
.7 
.8 
.9 




1 
1 
2 
2 
3 
3 
4 


i 

8 

2 

I 
S 


933 
934 
935 
936 
937 


988 
97 034 
081 
127 
174 


993 
039 
086 
132 
178 


997 
044 
090 
137 
183 


*002 
048 
095 
141 
188 


*007 
053 
099 
146 
192 


*011 
058 
104 
151 
197 


*016 
062 
109 
155 
202 


*026 
067 
113 
160 
206 


*025 
072 
118 
164 
211 


*030 
076 
123 
169 
215 


938 


220 


225 


229 


234 


239 


243 


248 


252 


257 


262 


939 


266 
313 


271 
317 


276 

322 


286 
326 


285 
33l 


289 
336 


294 


299 
345 


303 
349 


308 
354 


940 


340 




941 


359 


365 


368 


373 


377 


382 


386 


391 


396 


400 




942 


405 


409 


414 


419 


423 


428 


432 


437 


442 


446 




943 


451 


456 


466 


465 


469 


474 


479 


483 


488 


492 




944 


497 


502 


506 


511 


515 


520 


525 


529 


534 


538 




945 


543 


548 


552 


557 


561 


566 


570 


575 


580 


584 




946 


589 


593 


598 


603 


607 


612 


616 


621 


626 


630 




947 


635 


639 


644 


649 


653 


658 


662 


667 


671 


676 




948 


681 


685 


690 


694 


699 


703 


708 


713 


717 


722 




949 


726 


731 


736 


746 


745 


749 


754 


758 


763 


768 




950 


772 


777 
1 


781 
2 


786 


790 
4 


795 
5 


800 
6 


804 

7 


809 
8 


813 
9 




N. 





3 


P. P. 



598 







TABLE V.- 


-LOGAl.. 


THMS OF NUMBERS. 




N. 





1 
777 


2 

78l 


3 

786 


4 

790 


5 

795 


6 

800 


7 
804 


8 


9 


P. 


P. 


950 


97 772 


809 


813 






951 


818 


822 


827 


831 


836 


841 


845 


850 


854 


859 






952 


863 


868 


873 


877 


882 


886 


891 


895 


900 


904 






953 


909 


914 


918 


923 


927 


932 


936 


941 


945 


950 






954 


955 


959 


964 


968 


973 


977 


982 


986 


991 


996 






955 


98 000 


005 


009 


014 


018 


023 


027 


032 


036 


041 






956 


046 


050 


055 


059 


064 


068 


073 


077 


082 


086 




0^9 
1.0 

1.5 


957 
958 


091 
136 


095 
141 


100 
145 


105 
150 


109 
154 


114 
159 


118 
163 


123 
168 


127 
173 


132 
177 


.1 

.2 
.3 
.4 
.5 
.6 
.7 
• 8 
.9 


959 


182 

227 

272 
317 
362 
407 


186 

23l 

277 
322 
367 
412 


191 

236 

281 
326 
371 
416 


195 


200 


204 


209 


213 


218 


222 
268 


960 


240 

286 
331 
376 

421 


245 

290 
335 
380 

425 


249 


254 


259 


263 


2.0 . 

2.5 


961 
962 
963 
964 


295 
340 
385 
430 


299 
344 
389 
434 


304 
349 
394 
439 


308 
353 
398 
443 


313 
358 

403 
448 


3.0 
3.5 
4.0 
4.5 


965 


452 


457 


461 


466 


470 


475 


479 


484 


488 


493 






966 


497 


502 


506 


511 


515 


520 


524 


529 


533 


538 






967 


542 


547 


551 


556 


560 


565 


569 


574 


578 


58J 






968 


587 


592 


596 


601 


605 


610 


614 


619 


623 


628 






969 


632 


637 


641 


646 


650 


655 


659 


663 


668 


672 


.1 

.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 




970 


677 

722 


681 


686 


690 


695 


699 


704 


708 


713 


717 




971 


726 


731 


735 


740 


744 


749 


753 


757 


762 


0^ 
0.9 

2^2 

U 


972 

973' 

974 

975 

976 


766 
811 
856 
900 
945 


771 
815 
860 
905 
949 


775 
820 
865 
909 
954 


780 
824 
869 
914 
958 


784 
829 
873 
918 
963 


789 
833 
878 

922 
967 


793 
838 
882 
927 
971 


798 
842 
887 
931 
976 


802 
847 
891 
936 
980 


807 
851 
896 
940 
985 


977 


989 


994 


998 


*003 


^007 


=^011 


*016 


*020 


*025 


*029 


978 


99 034 


038 


043 


0-^7 


051 


056 


060 


065 


069 


074 


979 


078 


082 


087 


091 


096 


100 


105 


109 


113 


118 


980 


122 


127 


131 


136 


140 


145 


149 


153 


158 


162 




981 


167 


171 


176 


180 


184 


189 


193 


198 


202 


206 




982 


211 


215 


220 


224 


229 


233 


237 


242 


246 


251 






983 


255 


260 


264 


268 


273 


277 


282 


286 


290 


295 






984 


299 


304 


308 


312 


317 


321 


326 


330 


335 


339 






985 


343 


348 


352 


357 


36x 


365 


370 


374 


379 


383 






986 


387 


392 


396 


401 


405 


409 


414 


418 


423 


427 




4 
0.4 


987 


431 


436 


440 


445 


449 


453 


458 


462 


467 


471 


.1 
.2 
• 3 
.4 
.5 
.6 
.7 
.8 
.9 


988 


475 


480 


484 


489 


493 


497 


502 


506 


511 


515 


989 


519 


524 


528 


533 


537 


541 


546 


550 


554 


559 
603 


0.8 
1.2 


990 


563 
607 


568 
611 


572 


576 


581 


585 


590 


594 


598 


1.6 
2.0 


991 


616 


620 


625 


629 


633 


638 


642 


647 


2.4 
2.8 


992 


651 


655 


660 


664 


668 


673 


677 


682 


686 


690 


993 


695 


699 


703 


708 


712 


717 


721 


725 


730 


734 


3.2 
3.6 


994 


738 


743 


747 


751 


756 


760 


765 


769 


773 


778 


995 


782 


786 


791 


795 


800 


804 


808 


813 


817 


821 






996 


826 


830 


834 


839 


843 


847 


852 


856 


861 


865 






997 


869 


874 


878 


882 


887 


891 


895 


900 


904 


908 






998 


913 


917 


922 


926 


930 


935 


939 


943 


948 


952 






999 


956 


961 


965 


969 


974 


978 


982 


987 


991 


995 






1000 


00 000 


004 


008 


013 


017 


021 


026 


030 


034 


039 






N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P.] 


P. 



59y 







TABLE V - 


-LOGARITHMS OF NUMBERS. 










N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P. P. 


1000 


000 000 


043 


087 


130 


173 


217 


260 


304 


347 


390 




01 


434 


477 


521 


564 


607 


651 


694 


737 


781 


824 


02 


867 


911 


954 


997 


*041 


*084 


*127 


*171 


*214 


*257 




03 


001 301 


344 


387 


431 


474 


517 


560 


604 


647 


690 




04 


733 


777 


820 


863 


906 


950 


993 


*036 


*079 


*123 




05 


002 166 


209 


252 


295 


339 


382 


425 


468 


511 


555 




06 
07 
08 


598 

003 029 

460 


641 
072 
50.^ 


684 
115 
546 


727 
159 
590 


770 
202 
633 


814 
245 
676 


857 
288 
719 


900 
331 
762 


943 
374 
805 


986 
417 
848 


.1 
.2 
.3 


43 


43 

4.3 


09 


891 


934 


977 


*020 


*063 


*106 


*149 


*192 


*235 


*278 


13. 


5 


12 


9 


1010 


004 321 


364 


407 


450 


493 


536 


579 


622 


665 


708 


.4 
.5 


17. 
21. 


4 
7 


17 
21 


2 
5 


11 


751 


794 


837 


880 


923 


966 


*009 


*05l 


*094 


n37 


.6 
.7 
.8 
.9 


26 
30 
34 


1 
4 

-? 


25 
30 
34 


8 

1 
4 

•7 


12 


005 180 


223 


266 


309 


352 


395 


438 


481 


523 


566 


13 


609 


652 


695 


738 


781 


824 


866 


909 


952 


995 


14 


006 038 


081 


123 


166 


209 


252 


295 


337 


380 


423 






15 


466 


509 


551 


594 


637 


680 


722 


765 


808 


851 




16 


893 


936 


979 


*022 


*064 


*107 


*150 


*193 


*235 


*278 




17 


007 321 


363 


40C 


449 


491 


534 


577 


620 


662 


705 




18 


748 


790 


833 


875 


918 


961 


*003 


*046 


*089 


131 




19 


008 174 


217 


259 


302 


344 


387 


430 


472 


515 


557 




1030 


600 


642 


685 


728 


770 


813 


855 


898 


940 


983 




21 


009 025 


068 


111 


153 


196 


238 


281 


323 


366 


408 


.1 
.2 
.3 
.4 
.5 
.6 


f- 


42 

4.2 

8.4 

12 6 


22 


451 


493 


536 


578 


621 


663 


706 


748 


790 


833 


8 

12 
17 
21 


5 
7 
n 


23 


875 


918 


960 


*003 


*045 


*088 


*130 


*172 


*215 


*257 


24 


010 300 


342 


385 


427 


469 


512 


554 


.596 


639 


681 


16.8 
21.0 


25 


724 


766 


808 


851 


893 


935 


978 


*020 


*062 


*105 


i 


26 


Oil 147 


189 


232 


274 


316 


359 


401 


443 


486 


528 


25 


5 


25.2 


27 


570 


612 


655 


697 


739 


782 


824 


866 


908 


951 


.7 
.8 


29 

31 


7 



29.4 
33.6 


28 


993 


*035 


*077 


*120 


*162 


*204 


*246 


*288 


*331 


*373 


29 


012 415 


457 


500 


542 


584 


626 


668 


710 


753 


795 


.9 


38 


2 


37. 8 


1030 


837 


879 


921 


963 


*006 


*048 


*090 


*132 


174 


216 




31 


013 258 


301 


343 


385 


427 


469 


511 


553 


595 


637 




32 


679 


722 


764 


806 


848 


890 


932 


974 


*016 


*058 




33 


014 100 


142 


184 


226 


268 


310 


352 


394 


436 


478 




34 


520 


562 


604 


646 


688 


730 


772 


814 


856 


898 




35 


940 


982 


*024 


*066 


*108 


*150 


*192 


*234 


*276 


*318 




36 


015 360 


401 


443 


485 


527 


569 


611 


653 


695 


737 


aT At 


37 


779 


820 


862 


904 


946 


988 


*030 


*072 


*113 


155 


.1 


4 T 


4-1 


38 


016 197 


239 


281 


323 


364 


406 


448 


490 


532 


573 


• 2 


8 


3 


82 


39 


615 


657 


699 


741 


782 


824 


866 


908 


950 


991 


.3 


12 


.4 


12.3 


1040 


017 033 


075 


117 


158 


200 


242 


284 


325 


367 


409 


= 4 
.5 
.6 
.7 


16 
20 
24 
29 




16.4 
20.5 


41 


450 


492 


534 


576 


617 


659 


701 


742 


784 


826 


24-6 
28. 7 


42 


867 


909 


951 


992 


*034 


*076 


*117 


*159 


^201 


^242 


.8 


SS 


•2 
9 


32-8 


43 


018 284 


326 


367 


409 


451 


492 


534 


575 


J^'^ 


^^S^ 


.9 


<^7 


<IR.Q 


44 


700 


742 


783 


825 


867 


908 


950 


991 


-^O'S'S 


'*-UV4 




45 


019 116 


158 


199 


241 


282 


324 


365 


407 


448 


490 




46 


531 


573 


614 


656 


697 


739 


780 


822 


863 


90b 




47 


946 


988 


*029 


*071 


*112 


*154 


*195 


*237 


*278 


•^320 




48 


020 361 


402 


444 


485 


527 


568 


610 


651 


««^ 


734 




49 


775 


817 


858 


899 


941 


982 


*024 


*065 


*106 


•*-148 




1050 


021 189 


230 


272 
2 


313 
3 


354 

4 


396 
5 


437 
6 


478 

7 


520 
8 


56l 




N. 





1 


9 


P.P. 



6o0 



TABLE v.— LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P 


.P. 


1050 


021 


189 


230 


272 


313 


354 


396 


437 


478 


520 


56l 




fi 


51 




602 


644 


685 


726 


768 


809 


850 


892 


933 


974 


.1 


52 


022 


015 


057 


098 


139 


181 


222 


263 


304 


346 


387 


.2 


8 


3 


53 




428 


469 


511 


552 


593 


.634 


676 


717 


758 


799 


.3 


12 


4 


54 




840 


882 


923 


964 


*005 


*046 


*088 


*129 


*170 


*211 


.4 


16 


6 


55 


023 


252 


293 


335 


376 


417 


458 


499 


540 


581 


623 


.5 


20 


7 


56 




664 


705 


746 


787 


828 


869 


910 


951 


993 


*034 


.6 


24 


9 


57 


024 


075 


116 


157 


198 


239 


280 


321 


362 


403 


444 


•7 


29 





58 




485 


526 


568 


609 


650 


691 


732 


773 


814 


855 


.8 


33 


2 


59 


025 


896 


937 


978 


*019 


*060 


*101 


*142 


*183 


*224 


*265 


.9 


37 


3 


1060 


306 


347 


388 


429 


469 


510 


55l 


592 


633 


674 






























41 


61 




715 


756 


797 


838 


879 


920 


961 


*002 


*042 


*083 


.1 


4.1 


62 


026 


124 


165 


206 


247 


288 


329 


370 


410 


451 


492 


.2 


8 


2 


63 




533 


574 


615 


656 


696 


737 


778 


819 


860 


901 


• 3 


12 


3 


64 




941 


982 


*023 


*064 


*105 


*145 


*186 


^227 


*268 


*309 


.4 


16 


4 


65 


027 


349 


390 


431 


472 


512 


553 


594 


635 


675 


716 


.5 


20 


5 


66 




757 


798 


838 


879 


920 


961 


*001 


'^=042 


*083 


*123 


.6 


24 


6 


67 


028 


164 


205 


246 


286 


327 


368 


408 


449 


490 


530 


.7 


28 


7 


68 




571 


612 


652 


693 


734 


774 


815 


856 


896 


937 


.8 


32 


8 


69 


029 


977 


*018 


*059 


*099 


*140 


*181 


'*=22] 


*262 


*302 


*343 


.9 
.1 


36 


9 


1070 


384 


424 


465 


505 


546 


586 


627 


668 


708 


749 


t% 


71 


789 


830 


870 


911 


951 


992 


*032 


*073 


*114 


*154 


72 


030 


195 


235 


276 


316 


357 


397 


438 


478 


519 


559 


.2 


8 


1 


73 




599 


640 


680 


721 


761 


802 


842 


883 


923 


964 


.3 


12 


1 


74 


031 


004 


044 


085 


125 


166 


206 


247 


287 


327 


368 


.4 


16 


2 


75 




408 


449 


489 


529 


570 


610 


651 


691 


731 


772 


.5 


20 


2 


76 




812 


852 


893 


933 


973 


*014 


*054 


*094 


*135 


*175 


.6 


24 


3 


77 


032 


215 


256 


296 


336 


377 


417 


457 


498 


538 


578 


• 7 


28 


3 


78 




619 


659 


699 


739 


780 


820 


860 


900 


941 


981 


.8 


32 


4 


79 


033 


021 


061 


102 


142 


182 


222 


263 


303 


343 


383 


.9 


36 


4 


idso 




424 


464 


504 


544 


584 


625 


665 


705 


745 


785 


.1 


40 

4.0 


81 


825 


866 


906 


946 


986 


*026 


*066 


*107 


147 


187 


82 


034 


227 


267 


307 


347 


388 


428 


468 


508 


548 


588 


.2 


80 


83 




628 


668 


708 


748 


789 


829 


869 


909 


949 


989 


.3 


12.0 


84 


035 


029 


069 


109 


149 


189 


229 


269 


309 


349 


389 


.4 


16-0 


85 




429 


470 


510 


550 


590 


630 


670 


710 


750 


790 


.5 


20.0 


86 




830 


870 


910 


950 


990 


*029 


*069 


*109 


*149 


*189 


.6 


24.0 


87 


036 


229 


269 


309 


349 


389 


429 


469 


509 


549 


589 


.7 


28. 


88 




629 


669 


708 


748 


788 


828 


868 


908 


948 


988 


.8 


32.0 


89 


037 028 


068 


107 


147 


187 


227 


267 


307 


347 


386 


.9 


36.0 


1090 




426 


466 


506 


546 


586 


625 


665 


705 


745 


785 




39 

39 


91 




825 


864 


904 


944 


984 


*023 


*063 


*103 


143 


183 


.1 


92 


038 


222 


262 


302 


342 


381 


421 


461 


501 


540 


580 


.2 


7 


9 


93 




620 


660 


699 


739 


779 


819 


858 


898 


938 


977 


• 3 


11 


.8 


94 


039 


017 


057 


096 


136 


17f 


216 


255 


295 


335 


374 


.4 


15 


8 


95 




414 


454 


493 


533 


572 


612 


652 


691 


731 


771 


.5 


19 


.7 


96 




810 


850 


890 


929 


969 


*008 


*048 


*088 


*127 


*167 


.6 


23 


7 


97 


040 


206 


246 


286 


325 


365 


404 


444 


483 


523 


563 


.7 


27 


.6 


98 




602 


642 


681 


721 


760 


800 


839 


879 


918 


958 


.8 


31 


. 6 


99 


041 


997 
392 


*037 
432 


*076 
471 


*116 
511 


*155 
550 


*195 


*234 


*274 


*313 


*353 


.9 


35 


• 5 


1100 


590 
5 


629 
6 


669 

7 


708 


748 




N. 





1 


2 


3 


4 


8 


9 


P 


.P. 



601 



TABLE VT— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 



Log sin <j> = log 96" 4- S. 


- 


0° 


log <^" = log sin 


^ + S\ 


Log tan 4> = log 0'' + T. 




log^' 


' = log tan 4> + T\ 


n 


f 


S 


T I.C 


>§. Sin. 


W 


T' Log 


r. Tan, 








4.685 57 


57 


00 


5.314 42 


42 


— 00 


60 


1 


57 


57 6 


.46 372 


42 


42 6 


.46 37 


120 


2 


57 


57 


.76 475 


42 


42 


• 76 47 


180 


3 


57 


57 


.94 084 


42 


42 


.94 08 


240 


4 


57 


57 7 


.06 578 


42 


42 7 


06 57 


300 


5 


4.685 57 


57 7 


.16 269 


5.314 42 


42 7 


.16 261 


360 


6 


57 


57 


.24 187 


42 


42 


.24 18 


420 


7 


57 


57 


.30 882 


42 


42 


.30 88 


480 


8 


57 


57 


• 36 681 


42 


42 


.36 681 


540 


9 


57 


57 


.41 797 


42 


42 


• 41 79 


600 


10 


4.685 57 


57 7 


.46 372 


5.314 42 


42 7 


.46 37 


660 


11 


57 


57 


• 50 512 


42 


42 


.50 51! 


720 


12 


57 


57 


.54 290 


42 


42 


.54 29: 


780 


13 


57 


57 


.57 767 


42 


42 


.57 76 


840 


14 


57 


57 


.60 985 


42 


42 


• 60 98 


900 


15 


4-685 57 


58 7 


.63 981 


5,. 314 42 


42 7 


.63 981 


960 


16 


57 


58 


.66 784 


42 


42 


.66 785 


1020 


17 


57 


58 


.69 417 


42 


42 


.69 418 


1080 


18 


57 


58 


.71 899 


42 


42 


.71900 


1140 


19 


57 


58 


. 74 248 


42 


42 


• 74 248 


1200 


30 


4.685 57 


58 7 


.76 475 


5.314 43 


42 7 


.76 476 


1260 


21 


57 


58 


.78 594 


43 


42 


.78 595 


1320 


22 


57 


58 


.80 614 


43 


42 


.80 615 


1380 


23 


57 


58 


.82 545 


43 


42 


.82 546 


1440 


24 


57 


58 


.84 393 


43 


42 


.84 394 


1500 


25 


4.685 57 


58 7 


86 166 


5.314 43 


41 7 


86 167 


1560 


26 


57 


58 


87 869 


43 


41 


87 871 


1620 


27 


57 


58 


89 508 


43 


41 


89 510 


1680 


28 


57 


58 


91088 


43 


41 


91089 


1740 


29 


57 


58 


92 612 


43 


4l 


92 613 


1800 


30 


4.685 57 


58 7 


94 084 


5.314 43 


41 7 


94 086 


1860 


31 


57 


58 


95 508 


43 


41 


95 510 


1920 


32 


57 


58 


96 887 


43 


41 


96 889 


1980 


33 


57 


59 


98 223 


43 


41 


98 225 


2040 


34 


57 


59 


99 520 


43 


41 


99 522 


2100 


35 


4.685 56 


59 8 


00 778 


5.314 43 


41 8 


00 781 


2160 


36 


56 


59 


02 002 


43 


41 


02 004 


2220 


37 


56 


59 


03 192 


43 


41 


03 194 


2280 


38 


56 


59 


04 350 


43 


40 


04 352 


2340 


39 


56 


59 


05 478 


43 


40 


05 481 


2400 


40 


4.685 56 


59 8. 


06 577 


5.314 43 


40 8. 


06 580 


2460 


41 


56 


59 


07 650 


43 


40 


07 653 


2520 


42 


56 


59 


08 696 


43 


40 


08 699 


2580 


43 


56 


60 


09 718 


43 


40 


09 721 


2640 


44 


56 


60 


10 716 


43 


40 


10 720 


2700 


45 


4.685 56 


60 8 


11 692 


5.314 44 


40 8. 


11 696 


2760 


46 


56 


60 


12 647 


44 


40 


12 651 


2820 


47 


56 


60 


13 581 


44 


40 


13 585 


2880 


48 


56 


60 


14 495 


44 


39 


14 499 


2940 


49 


56 


60 


15 390 


44 


39 


15 395 


3000 


50 


4.685 56 


60 8 


16 268 


5.314 44 


39 8. 


16 272 


3060 


51 


56 


60 


17 128 


44 


39 


17 133 


3120 


52 


56 


61 


17 971 


44 


39 


17 976 


3180 


53 


56 


61 


18 798 


44 


39 


18 803 


3240 


54 
55 


55 


61 


19 610 


44 


39 


19 615 


3300 


4-685 55 


61 8 


20 407 


5.314 44 


39 8. 


20 412 


3360 


56 


55 


6: 


21 189 


44 


38 


21 195 


3420 


57 


55 


6: 


21958 


44 


38 


21964 


3480 


58 


55 


6 


22 713 


44 


38 


22 719 


3540 


59 


55 


62 


23 455 


44 


38 


23 462 



602 



TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES. 



Log sin (f> = log 4)'' + S. 




r 


log9£>' 


= log 


sin d> + S\ 


Log tan = log 0" + T. 




log 4>' 


= log tan + T\ 


ff 


# 


S 


T 


I.og. Sin. 


S' 


T' 


Log. Tan. 


3600 





4.685 55 


62 


8.24 185 


5.314 44 


38 


8.24 192 


3660 


1 


55 


62 


.24 903 


45 


38 


-24 910 


3720 


2 


55 


62 


.25 609 


45 


38 


-25 616 


3780 


3 


55 


62 


.26 304 


45 


37 


-26 311 


3840 


4 


55 


62 


.26 988 


45 


37 


• 26 995 


3900 


5 


4.685 55 


62 


8-27 661 


5-314 45 


37 


8-27 669 


3960 


6 


55 


63 


.28 324 


45 


37 


-28 332 


4020 


7 


54 


63 


.28 977 


45 


37 


-28 985 


4080 


8 


54 


63 


.29 620 


45 


37 


-29 629 


4140 


9 


54 


63 


.30 254 


45 


36 


-30 263 


4200 


10 


4.685 54 


63 


8.30 879 


5.31445 


36 


8-30 88§ 


4260 


11 


54 


63 


.31 495 


45 


36 


.31 504 


4320 


12 


54 


64 


.32 102 


45 


36 


.32 112 


4380 


13 


54 


64 


.32 701 


46 


36 


.32 711 


^440 


14 


54 


64 


.33 292 


46 


36 


.33 302 


4500 


15 


4.685 54 


64 


8-33 875 


5.314 46 


35 


8. 33 885 


4560 


16 


54 


64 


.34 450 


46 


35 


.34 461 


4620 


17 


54 


65 


.35 018 


46 


35 


.35 029 


4680 


18 


54 


65 


.35 578 


46 


35 


-35 589 


4740 


19 


53 


65 


.36 131 


46 


35 


-36 143 


4800 


30 


4. 685 53 


65 


8.36 677 


5.314 46 


34 


8-36 689 


4860 


21 


53 


65 


.37 217 


46 


34 


.37 229 


4920 


22 


53 


65 


.37 750 


46 


34 


.37 762 


4980 


23 


53 


66 


.38 276 


46 


34 


• 38 289 


5040 


24 


53 


66 


.38 796 


47 


34 


-38 809 


5100 


25 


4.685 53 


66 


8.39 310 


5.314 47 


33 


8-39 323 


5160 


26 


53 


66 


.39 818 


47 


33 


-39 83l 


5220 


27 


53 


67 


.40 320 


47 


33 


.40 334 


5280 


28 


52 


67 


.40 816 


47 


33 


.40 830 


5340 


29 


52 


67 


' .41 307 


47 


33 


-41 32l 


5400 


30 


4.685 52 


67 


8.41 792 


5-314 47 


32 


8-41 807 


5460 


31 


52 


67 


.42 271 


47 


32 


.42 287 


5520 


32 


52 


68 


.42 746 


47 


32 


.42 762 


5580 


33 


52 


68 


.43 215 


48 


32 


.43 231 


5640 


34 


52 


68 


.43 680 


48 


3l 


-43 696 


5700 


35 


4.685 52 


68 


8-44 139 


5-314 48 


31 


8-44 156 


5760 


36 


52 


69 


.44594 


48 


31 


.44 611 


5820 


37 


51 


69 


.45 044 


48 


31 


.45 061 


5880 


38 


51 


69 


.45 489 


48 


30 


.45 507 


5940 


39 


51 


69 


.45 930 


48 


30 


.45 948 


6000 


40 


4.685 51 


69 


8-46 366 


5.314 48 


30 


846 385 


6060 


41 


51 


70 


.46 798 


49 


30 


.46 817 


6120 


42 


51 


70 


.47 226 


49 


30 


.47 245 


6180 


43 


51 


70 


.47 650 


49 


29 


.47 669 


6240 


44 


51 


70 


-48 069 


49 


29 


-48 089 


6300 


45 


4-685 50 


71 


8. 48 485 


5.31449 


29 


8- 48 505 


6360 


46 


50 


71 


.48 896 


49 


28 


.48 917 


6420 


47 


50 


71 


-49 304 


49 


28 


.49 325 


6480 


48 


50 


72 


-49 708 


49 


28 


.49 729 


6540 


49 


50 


72 


• 50 108 


50 


28 


-50 130 


6600 


50 


4.685 50 


72 


8-50 504 


5^314 50 


27 


8-50 526 


6660 


51 


50 


72 


-50 897 


50 


27 


.50 920 


6720 


52 


50 


73 


-51 286 


50 


27 


.51 310 


6780 


53 


49 


73 


-51 672 


50 


27 


.51 696 


6840 


54 


49 


73 


• 52 055 


50 
5.314 50 


26 
26 


-52 079 


6900 


55 


4.685 49 


73 


8-52 434 


8-52 458 


6960 


56 


49 


7^ 


.52 810 


51 


26 


.52 835 


7020 


57 


49 


7< 


.53 183 


51 


25 


.53 208 


7080 


58 


49 


7; 


.53 552 


51 


25 


.53 578 


7140 


59 


49 


75 


.53 918 


51 


25 


.53944 



603 



TABLE VI.— LOGARITHMIC SINES AND TANGENTS OF SMALL ANGLES 



Log sin (f> == log <f>" + S, 




2° 


log <j)' 


' = log 


sin d> + S', 


Log tan 4> = log ^" + T. 




log <})' 


= log tan ^ + T\ 


tf 


/ 


S 


T 


Log, Sin. 


S' 


T' 


I^og. Tan, 


7200 





4.685 48 


75 


8.54 282 


5.314 51 


25 


8. 54 308 


7260 


1 


48 


75 


.54 642 


5l 


24 


.54 669 


7320 


2 


48 


75 


.54 999 


51 


24 


.55 027 


7380 


3 


48 


76 


.55 354 


52 


24 


.55 381 


7440 


4 
5 


48 


76 


• 55 705 


52 


23 


.55 733 


7500 


4.685 48 


76 


8.56 054 


5.314 52 


23 


8.56 083 


7560 


6 


48 


7Z 


.56 400 


52 


23 


.56 429 


7620 


7 


47 


77 


.56 743 


52 


22 


.56 772 


7680 


8 


47 


77 


.57 083 


52 


22 


.57 113 


7740 


9 


47 


78 


.57 421 


52 


22 


.57 452 


7800 


10 


4.685 47 


78 


8.57 756 


5.314 53 


22 


8.57 787 


7860 


11 


47 


78 


.58 089 


53 


2l 


.58 121 


7920 


12 


47 


79 


.58 419 


53 


21 


.58 451 


7980 


13 


46 


79 


.58 747 


53 


21 


.58 779 


8040 


14 


46 


79 


.59 072 


53 


20 


.59 105 


8100 


15 


4.685 46 


80 


8.59 395 


5.314 53 


20 


8.59 428 


8160 


16 


46 


80 


.59 715 


54 


20 


.59 749 


8220 


17 


46 


80 


.60 033 


54 


19 


.60 067 


8280 


18 


46 


81 


.60 349 


54 


19 


.60 384 


8340 


19 


45 


81 


.60 662 


54 


19 


.60 698 


8400 


20 


4.685 45 


81 


8.60 973 


5.314 54 


18 


8.61009 


8460 


21 


45 


82 


.61 282 


54 


18 


.61319 


8520 


22 


45 


82 


.61 589 


55 


18 


.61 626 


8580 


23 


45 


82 


.61 893 


55 


17 


.61 931 


8640 


24 


45 


83 


• 62 196 


55 


17 


.62 234 


8700 


25 


4.685 44 


83 


8.62 496 


5.314 55 


16 


8.62 535 


8760 


26 




83 


.62 795 


55 


16 


.62 834 


8820 


27 


44 


8^: 


.63 091 


55 


16 


.63 13] 


8880 


28 


44 


8^i 


.63 385 


56 


]5 


.63 425 


sgTo 


29 


44 


84 


.63 677 


56 


15 


.63 718 


9000 


30 


4.685 43 


85 


8.63 968 


5.314 56 


15 


8-64 009 


9060 


31 


43 


85 


.64 256 


56 


14 


.64 298 


9120 


32 




86 


.64 543 


56 


14 


.64 585 


9180 


33 


43 


86 


.64 827 


57 


14 


.64 870 


9240 


34 
35 


43 


86 


.65 110 


57 


13 


.65 153 


9300 


4.685 43 


87 


8.65 391 


5.314 57 


13 


8.65 435 


9360 


36 


42 


87 


.65 670 


57 


12 


.65 715 


9420 


37 


42 


87 


.65 947 


57 


12 


.65 993 


9480 


38 


42 


88 


.66 223 


58 


12 


.66 269 


9540 


39 


42 


88 


.66 497 


58 


ll 


.66 543 


9600 


40 


4.685 42 


89 


8.66 769 


5.314 58 


11 


8.66 816 


9660 


41 


41 


89 


.67 039 


58 


10 


.67 087 


9720 


42 


4l 


89 


.67 308 


58 


10 


.67 356 


9780 


43 


41 


90 


.67 575 


59 


10 


.67 624 


9840 


44 


41 


90 


.67 840 


59 


09 


.67 890 


9900 


45 


4.685 41 


91 


8.68 104 


5.314 59 


09 


8.68 154 


9960 


46 


40 


91 


.68 360 


59 


08 


.68 417 


10020 


47 


40 


9l 


.68 627 


59 


08 


.68 678 


10080 


48 


40 


92 


.68 886 


60 


08 


.68 938 


10140 


49 


40 


92 


.69 144 


60 


07 


.69 196 


10200 


50 


4.685 40 


93 


8.69 400 


5.314 60 


07 


8.69 453 


10260 


51 


39 


93 


.69 6§4 


60 


06 


.69 708 


10320 


52 


39 


93 


.69 907 


60 


06 


« 69 961 


10380 


53 


39 


94 


.70 159 


SI 


06 


.70 214 


10440 


54 
55 


39 


94 


• 70 409 


61 


05 


.70 464 


10500 


4.685 38 


95 


8.70 657 


5.314 61 


05 


8.70 714 


10560 


56 


38 


95 


.70 905 


6:. 


04 


.70 962 


10620 


57 


38 


96 


.71 150 


6:. 


04 


.71 208 


10680 


58 


38 


96 


.71 395 


62 


03 


.71453 


10740 


59 


38 


97 


.71 638 


62 


03 


.71 697 



604 



TABLE VII— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



r 


Log. Sin. 


D 


Log. Tan. 


Com. D. 


Log. Cot. 


Log. Cos. 






1 

2 
3 
4 


6 
6 
6 

7 


— 00 

46 372 
76 475 
94 084 
06 578 


30103 
17609 
12494 
9691 
7918 
6695 
5799 
5115 
4575 
4139 
3778 
3476 
3218 
2996 
2803 
2633 
2482 
2348 

2227 
2119 
2020 
1930 
1848 
1772 
1703 
1639 
1579 
1524 
1472 
1424 
1379 
1336 
1296 
1258 
1223 
1190 
1158 
1128 
1099 
1072 
1046 
1022 
998 
976 
954 
934 
914 
895 

877 
860 
843 
827 
811 
797 
782 
768 
755 
742 
730 


6 
6 
6 
7 


-00 
46 372 
76 475 
94 084 
06 578 


30103 
17609 
12494 
9691 
7918 
6694 
5799 
5115 
4575 
4139 
3779 
3476 
3218 
2996 
2803 
2633 
2482 
2348 

2227 
2119 
2020 
1930 
1848 

1773 
1703 
1639 
1579 
1524 
1472 
1424 
1379 
1336 
1296 

1259 
1223 
1190' 
1158 
1128 
1099 
1072 
1046 
1022 
999 

976 
954 
934 
914 
895 
877 
860 
843 
827 
812 
797 
783 
768 
755 
742 
730 


+ 00 
3 53 627 
3^23 524 
3. 05 915 
2.93 421 









00 000 
00 000 
00 000 
00 000 
00 000 


60 

59 
58 
57 
5b 


5 
6 
7 
8 
9 


7 
7 
7 
7 
7 


16 269 
24 187 
30 882 
36 681 
41 797 


7 
7 
7 
7 
7 


16 269 
24 188 
30 882 
36 68l 
.41 797 


2 •83 730 
2.75 812- 
2.69 117 
2.63 318 
2.58 203 









.00 000 
00 000 

.00 000 
00 000 
00 000 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


7 
7 
7 
7 
7 


46 372 
50 512 
54 290 
57 767 
60 985 


7 
7 
7 
7 
7 


.46 372 
.50 512 
.54 291 
.57 767 
• 60 985 


2.53 627 
2.49 488 
2.45 709 
2.42 233 
2.39 014 




9 
9 
9 


00 000 
00 000 
99 999 
99 999 
99 999 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


7 
7 
7 
7 
7 


63 981 
66 784 
69 417 
71 899 
74 248 


7 
7 
7 
7 

7 


• 63 982 
.66 785 
.69 418 
.71900 
. 74 248 


2-36 018 
2-33 215 
2.30 582 
2.28 099 
2.25 75l 


9 
9 
9 
9 
9 


99 999 
99 999 
99 999 
99 999 
99 999 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


7 
7 
7 
7 
7 


76 475 
78 594 
80 614 
82 545 
84 393 


7 
7 
7 
7 
7 


.76 476 
.78 595 
.80 615 

• 82 546 

• 84 394 


2.23 524 
2.21405 
2.19 384 
2.17 454 
2.15 605 


9 
9 
9 
9 
9 


99 999 
99 999 
99 999 
99 999 
99 999 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


7 
7 
7 
7 
7 


86 166 

87 869 
89 508 

91 088 

92 612 


7 
7 
7 
7 

7 


• 86167 

• 87 871 

• 89 510 

• 91 089 

• 92 613 


2.13 832 
2.12 129 
2.10 490 
2.08 910 
2.07 386 


9 
9 
9 
9 
9 


99 999 
99 999 
99 998 
99 998 
99 998 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


7 
7 
7 
7 
7 


94 084 

95 508 

96 887 

98 223 

99 520 


7 
7 
7 
7 
7 


• 94 086 

• 95 510 

• 96 889 

• 98 225 

• 99 522 


2.05 914 
2.04 490 
2.03 111 
2'. 01 774 
2.00 478 


9 
9 
9 
9 
9 


99 998 
99 998 
99 998 
99 998 
99 998 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


8 
8 
8 
8 
8 


00 778 

02 002 

03 192 

04 350 

05 478 


8 
8 
8 
8 
8 


• 00 781 

• 02 004 

• 03 194 

04 352 

05 481 


1.99 219 
1.97 995 
1.96 805 
1-95 647 
1.94 519 


9 
9 
9 
9 
9 


99 997 
99 997 
99 997 
99 997 
99 997 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


8 
8 
8 
8 
8 


06 577 

07 650 

08 696 

09 718 

10 716 


8 
8 
8 
8 
8 


06 580 

07 653 

08 699 

09 72l 

10 720 


1.93 419 
1.92 347 
1.91300 
1.90 278 
1.89 279 


9 
9 
9 
9 
9 


99 997 
99 997 
99 997 
99 996 
99 996 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


8 
8 
8 
8 
8 


11 692 

12 647 

13 581 

14 495 

15 390 


8 
8 
8 
8 
8 


11 696 

12 651 

13 585 

14 499 

15 395 


1.88 303 
1.87 349 
1.86 415 
1^85 500 
1^84 605 


9 
9 
9 
9 
9 


99 996 
99 996 
99 996 
99 996 
99 995 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


8 
8 
8 
8 
8 


16 268 

17 128 

17 971 

18 798 

19 610 


8 
8 
8 
8 
8 


16 272 

17 133 

17 976 

18 803 

19 615 


1-83 727 
1^82 867 
1.82 023 
1.81 196 
1.80 384 


9 
9 
9 
9 
9 


99 995 
99 995 
99 995 
99 995 
99 994 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


8 
8 
8 
8 
8 


20 407 

21 189 

21 958 

22 713 

23 455 


8. 
8. 
8. 
8. 
8. 


20 412 

21 195 

21 964 

22 719 

23 462 


1.79 587 
1.78 804 
1.78 036 
1.77 280 
1.76 538 


9 
9 
9 
9 
9 


99 994 
99 994 
99 994 
99 994 
99 993 


5 
4 
3 
2 

1 


60 


8 


24 185 


8. 


24 192 


1-75 808 


9 


99 993 







Log. Cos. 


D 


Log. Cot. 1 


Com. D. 


Log. Tan. 


Log. Sin. 


' 



90° 



605 



89° 



TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS, 
AND COTANGENTS. 



178** 



log. Sin. 



8.24 185 
8.24 903 
8-25 609 
8.26 304 
8-26 988 



8.27 661 

8.28 324 

8. 28 977 

8.29 620 
8-30 254 



8. 30 879 
8-31495 
8.32 102 

8.32 701 
8-33 292 

8.33 875 
8 • 34 450 
8.35 018 
8-35 578 
8-36 131 



8. 36 677 

8.37 217 

8.37 750 

8.38 276 
8-38 796 

8.39 310 

8. 39 818 

8.40 320 

8.40 816 

8.41 307 



8.41 792 

8.42 271 

8.42 746 

8.43 215 
8-43 680 



8.44 139 
8-44 594 
8-45 044 

8.45 489 
845 930 



'8.46 366 

8.46 798 

8.47 226 

8.47 650 

8.48 069 



8. 48 485 

8.48 896 

8.49 304 
8.49 708 
8-50 108 



8.50 504 

8.50 897 

8.51 286 

8.51 672 

8.52 055 



8.52 434 

8.52 810 

8.53 183 
8. 53 552 
8. 53 918 



8-54 282 
Log. Cos. 



718 
706 
694 
684 
673 
663 
653 
643 
634 
625 
616 
607 
599 
591 

583 
575 
567 
560 
553 
546 
539 
533 
526 
520 
514 
508 
502 
496 
491 
48§ 
479 
474 
469 
464 

459 
454 
450 
445 
440 
436 
432 
428 
423 
419 
415 
411 
407 
404 
400 
396 
393 
389 
386 
382 
879 
375 
373 
369 
366 
363 



Log. Tan. 



8.24 192 

8.24 910 

8.25 616 

8.26 311 
8-26 995 



8.27 669 

8.28 332 
8.28 985 
8-29 629 
8-30 263 



8-30 888 
8-31504 
8-32 112 
832 711 
8. 33 302 



8. 33 885 

8.34 461 

8.35 029 
8-35 589 
8-36 143 



8. 36 689 

8.37 229 

8.37 762 

8.38 289 
8 38 809 



8.39 323 

8.39 831 

8.40 334 
8.40 830 
8.41321 



8-41807 
8-42 287 
8-42 762 
8.43 231 
8-43 696 



Com. D. 



8.44 156 

8.44 611 

8. 45 061 
8.45 507 
8-45 948 



8.46 385 
8-46 817 
8-47 245 
8-47 669 
8. 48 089 



8.48 505 
8-48 917 

8.49 325 
8.49 729 
8-50 130 



8.50 f 26 

8.50 120 

8.51 310 

8.51 696 

8.52 079 



8.52 458 

8.52 835 

8.53 208 
8.53 578 

_8^53_9_44_ 
8-54 308 
Log. Cot. 



718 

706 

695 

684 

673 

663 

653 

643 

634 

625 

616 

607 

599 

591 

583 

575 

568 

560 

553 

546 

539 

533 

527 

520 

514 

508 

502 

496 

491 

485 

480 

475 

469 

464 

460 

455 

450 

445 

441 

437 

432 

428 

424 

419 

416 

412 

408 

404 

400 

396 

393 

390 

386 

383 

379 

376 

373 

370 

366 

364 

Com. D. 



Log. Cot, 



1-75 808 
1-75 090 
1.74 383 
1-73 688 
1 . 73 004 



1-72 331 
1.71 667 
1.71014 
1.70 371 
1.69 736 



1.69 111 
1.68 495 
1.67 888 
1.67 288 
1.66 697 



1.66 114 
1.65 539 
1.64 971 
1.64 410 
1.63 857 



1.63 310 
1.62 771 
1.62 238 
1.61711 
1-61 191 



1-60 676 
1.60 168 
1.59 666 
1.59 169 
1.58 678 



1-58 193 
1-57 713 
1.57 238 
1-56 768 
1.56 304 



1.55 844 
1.55 389 
1.54 938 
1.54 493 
1.54 052 



1.53 615 
1.53 183 
1.52 754 
1-52 330 
1.51 911 



1-51495 
1-51083 
1-50 675 
1.50 270 
1.49 870 



1.49 473 
1.49 080 
1-48 690 
1-48 304 
1-47 921 



1-47 541 
1-47 165 
1.46 792 
1-46 422 
1.46 055 



1-45 691 
Log. Tan. 



Log, Cos. 



9.99 993 
9-99 993 
9-99 993 
9-99 992 
9.99 992 



9-99 992 
9.99 992 
9-99 992 
9-99 99l 
9-99 991 



9-99 991 
9.99 990 
9-99 990 
9-99 990 
9. 99 990 



9-99 989 
9-99 989 
9-99 989 
9-99 989 
9-99 988 



9-99 988 
9-99 988 
9-99 987 
9.99 987 
9-99 987 



9.99 986 
9-99 986 
9-99 986 
9-99 986 
9.99 985 



9.99 985 
9.99 985 
9.99 984 
9.99 984 
9.99 984 



9-99 983 
9-99 983 
9-99 982 
9.99 982 
9.99 982 



9-99 981 
9-99 981 
9-99 981 
9.99 980 
9-99 980 



9.99 979 
9.99 979 
9-99 979 
9-99 978 
9. 99 978 



9.99 978 
9.99 977 
9.99 977 
9-99 976 
9.99 976 



9.99 975 
9.99 975 
9.99 975 
9.99 974 
9-99 974 



9-99 973 
Log, Sin. 



91« 



606 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



177° 



Log. Sin, 



8. 54 282 
8-54 642 

8.54 999 

8.55 354 
8.55 705 



8.59 395 

8.59 715 

8.60 033 
8.60 349 
8.60 662 



8.56 054 
8.56 400 

8.56 743 

8.57 083 
8.57 421 



8.57 756 

8.58 039 
8.58 419 

8.58 747 

8.59 072 



8.60 973 

8.61 282 
8.61 589 

8.61 893 

8.62 196 



8.62 496 

8.62 795 

8.63 091 
8.63 385 
8.63 677 



8.63 968 

8.64 256 
8.64 543 

8.64 827 

8.65 110 



8.65 391 
8.65 670 

8.65 947 

8.66 223 
8-66 497, 



8.66 769 

8.67 039 
8.67 308 
8-67 575 
8.67 840 



8.68 104 
8.68 366 
8.68 627 

8.68 886 

8.69 144 



8.69 400 
8-69 654 
8-69 907 

8.70 159 
8.70 409 



8.70 657 

8.70 905 

8.71 150 
8-71 395 
8-71 638 
8-71 880 
Log. Cos. 



360 
357 
354 
35l 

348 
346 
343 
340 
338 
335 
332 
330 
327 
325 
323 
320 
318 
316 
313 

311 

309 

306 

304 

302 

300 

298 

296 

294 

292 

290 

288 

286 

284 

282 

281 

279 

277 

275 

274 

272 

270 

268 

267 

265 

264 

262 

260 

259 

257 

256 

254 

253 

251 

250 

248 

247 

245 

244 

243 

24l 



Log, Tan. 



8 


59 105 


8 


59 428 


8 


59 749 


8 


60 067 


8 


60 384 


8 


60 698 


8 


61 009 


8 


61 319 


8 


61 626 


8 


61 931 


8 


62 234 



54 308 

54 669 

55 027 
55 381 
55 733 



56 083 
56 429 

56 772 

57 113 
57 452 



57 787 

58 121 
58 451 
58 779 



62 535 

62 834 

63 131 
63 425 
63 71^ 



64 009 
64 298 
64 585 

64 870 

65 153 



65 435 
65 715 

65 993 

66 269 
66 543 



66 816 

67 087 
67 356 
67 624 
67 890 



68 154 
68 417 
68 678 

68 938 

69 196 



69 453 
69 708 

69 961 

70 214 
70 464 



70 714 

70 962 

71 208 
71 453 
71 697 



8-71 939 
Log. Cot. 



Com. D. 



360 
358 
354 
352 

349 
346 
343 
341 
338 

335 
333 
330 
328 
325 
323 
320 
318 
316 
314 

311 
309 
307 
305 
303 
300 
299 
297 
294 
293 
291 
288 
287 
285 
283 
28l 
280 
278 
276 
274 
272 
271 
269 
267 
266 
264 
262 
26l 
259 
258 

256 
255 
253 
252 
250 
249 
248 
246 
245 
243 
242 

Com. D. 



Log. Cot. 



45 691 
45 331 
44 973 
44 618 
44268 



43 917 
43 571 
43 227 
42 886 
42 548 



42 212 
41 879 
41 548 
41 220 
40 895 



40 571 
40 251 
39 932 
39 616 
39 302 



38 990 
38 681 
38 374 
38 068 
37 765 



37 465 
37 166 
36 869 
36 574 
36 281 



35 990 
35 702 
35 414 
35 129 
34 846 



34 565 
34 285 
34 007 
33 731 
33 456 



33 184 
32 913 
32 643 
32 376 
32 110 



31 845 
31 583 
31 32l 
31 062 
30 803 



30 547 
30 292 
30 038 
29 786 
29 535 



29 286 
29 038 
28 791 
28 546 
28 303 



28 060 



Log. Tan. 



Log. Cos. 



9-99 973 
9-99 973 
9-99 972 
9-99 972 
9. 99 971 



9-99 971 
9-99 971 
9.99 970 
9.99 970 
9-99 969 



9.99 969 
9.99 968 
9.99 968 
9.99 967 
9-99 967 



9.99 966 
9.99 966 
9.99 965 
9.99 965 
9.99 964 



9.99 964 
9.99 963 
9.99 963 
9.99 962 
9-99 962 



9.99 961 
9.99 961 
9.99 960 
9.99 959 
9-99 959 



9.99 958 
9.99 958 
9.99 957 
9.99 957 
9.99 956 



9.99 956 
9.99 955 
9.99 954 
9.99 954 
P. 99 953 



9-99 953 
9-99 952 
9-99 952 
9-99 951 
9.99 950 



9-99 950 
9-99 949 
9-99 948 
9-99 948 
9-99 947 



9-99 947 
9-99 946 
9- 99 945 
9-99 945 
9.99 944 



9-99 943 
9-99 943 
9-99 942 
9-99 942 
9-99 941 



9-99 940 
Log. Sin. 



607 



87' 



TABLE VII.— LOGARITHMIC SINES, COSINES, T.tTOENTS. 
AND COTANGENTS. 



176 



Log. Sin 



8-71 

• 72 
.72 

• 72 
.72 



880 
120 
359 
597 
833 
069 
302 
535 
766 
997 
226 
453 
680 
905 
129 
353 
574 
795 
015 
233 
451 
667 
883 
097 
310 

522 
733 
943 
152 
360 
567 
773 
978 
183 
386 
588 
789 
989 
189 
387 
585 
782 
977 
172 
366 
560 
752 
943 
134 
324 
513 
701 
888 
075 
260 
445 
629 
813 
995 
177 
358 



Log. Cos 



240 
239 
237 
236 
235 
233 
233 
231 
230 
229 
227 
226 
225 
224 
223 
221 
221 
219 
218 

217 
216 
215 
214 
213 
21§ 
211 
210 
209 
208 
207 
206 
205 
204 
203 
202 
20l 
200 
199 
198 
197 
197 
195 
195 
194 
193 
192 
191 
191 
189 
189 
188 
187 
186 
185 
185 
184 
183 
182 
182 
181 



Log. Tan. c.d. Log. Cot. 



8. 84 
Log. 



180 
420 
659 
896 
131 
366 
599 
831 
062 
292 
520 
748 
974 
199 
422 
645 
867 
087 
306 
524 
741 
958 
172 
386 
599 
8ll 
022 
232 
441 
648 
855 
061 
266 
470 
673 
875 
076 
276 
476 
674 
871 
068 
264 
459 
653 
846 
038 
230 
420 
610 
799 
987 
175 
361 
547 
732 
916 
100 
282 
464 
Cot, 



241 
240 
238 
237 
235 
235 
233 
232 
231 
229 
228 
227 
226 
225 
223 
223 
22l 
220 
219 
218 
217 
216 
214 
214 

213 
212 
210 
210 
209 
207 
207 
206 
204 
204 
203 
202 
20l 
200 
199 
198 
197 
197 
195 
195 
194 
193 
192 
191, 
190 
190 
188 
188 
187 
186 
185 
185 
184 
183 
182 
182 



1.24 577 
1.24 354 
1.24 133 
1.23 913 
1.23 693 



1.28 060 
1.27 819 
1.27 579 
1.27 341 
1.27 104 



1.26 868 
1.26 633 
1.26 400 
1.26 168 
1.25 037 



1.25 708 
1.25 479 
1.25 252 
1.25 026 
1.24 801 



1.23 475 
1.23 258 
1.23 042 
1.22 827 
1-22 613 



1.22 400 
1.22 188 
1.21 978 
1.21 768 
1.21 559 



1.21351 
1.21 144 
1.20 938 
1.20 734 
1.20 530 



1.20 327 
1.20 125 
1.19 923 
1.19 723 
1.19 524 




1.17 389 
1.17 201 
1.17012 
1.16 825 
1.16 638 



1.16 453 
1.16 268 
1.16 083 
1.15 900 
1.15 717 



c.d, 



1-15 535 



Log. Cos. 



99 940 
99 940 
99 939 
99 938 
99 938 



99 937 
99 936 
99 935 
99 935 
99 934 



99 933 
99 933 
99 932 
99 931 
99 931 



99 930 
99 929 
99 928 
99 928 
99 927 



99 926 
99 925 
99 925 
99 924 
99 923 



99 922 
99 922 
99 921 
99 920 
99 919 



99 919 
99 918 
99 917 
99 916 
99 916 



99 915 
99 914 
99 913 
99 912 
99 912 



99 911 
99 910 
99 909 
99 908 
99 907 



99 907 
99 906 
99 905 
99 904 
99903 



99 902 
99 902 
99 901 
99 900 
99 899 



99 898 
99 897 
99 896 
99 896 
99 895 



99 894 



93° 



Log, Tan. Log. Sin, 
608 



P. P. 





330 


330 


310 


6 


33.0 


82.0 


31.0 


7 


38.5 


37.3 


36.1 


8 


44.0 


42.6 


41-3 


9 


49.5 


48.0 


46.5 


10 


55.0 


53.3 


51.6 


20 


110.0 


106.6 


103.3 


30 


165.0 


160.0 


155.0 


40 


220.0 


213.3 


206.6 


50 


275.0 


266.6 


258.3 





290 


380 


870 


6 


29.0 


28.0 


27.0 


7 


33.8 


32.6 


31.5 


8 


38.6 


37.3 


36.0 


9 


43.5 


42.0 


40.5 


10 


48.3 


46.6 


45.0 


20 


96.6 


93.3 


90.0 


30 


145.0 


140.0 


135.0 


40 


193.3 


186.6 


180.0 


50 


241.6 


233.3 


225.0 





250 


240 


230 


6 


25.0 


24.0 


23.0 


7 


29.1 


28.0 


26.8 


8 


33.3 


32.0 


30-6 


9 


37.5 


36.0 


34.5 


10 


41.6 


40.0 


38.3 


20 


83.3 


80.0 


76.6 


30 


125.0 


120.0 


115.0 


40 


166.6 


160.0 


153.3 


50 


208.3 


200.0 


191.6 



210 

21.0 

24.5 

28.0 

31.5 

35.0 

70.0 

105.0 

140.0 

175.0 



200 

20.0 

23.3 

26.6 

30.0 

33.3 

66.6 

100.0 

133.3 

166.6 



190 

19.0 
22.1 
25.3 
28.5 
31.6 
63.3 
95.0 
126.6 
158.3 





9 


9 


8 


7 


6 


6 


0.9 


0.9 


0.8 


0.7 


0.6 


7 


1.1 


1.0 


0.9 


0.8 


0.7 


8 


1.2 


1.2 


1.0 


0.9 


0.8 


9 


1.4 


1.3 


1.2 


1.0 


0.9 


10 


1.6 


1.5 


1.3 


1.1 


1.0 


20 


3.1 


3.0 


2.6 


2.3 


2.0 


30 


4.7 


4.5 


4.0 


3.5 


3.0 


40 


6.3 


6.0 


5.3 


4.6 


4.0 
5.0 


50 


7.9 


7.5 


6.6 


5.8 



300 

30.0 

35.0 

40.0: 

45.0 

50.0 

100.0 

150.0, 

200.0 

250.0 

260 

26.0 

30.3 

34.6 

39.01 

43.3 j 

86.6 

130.01 

173.31 

216.6 

230 

22.0: 

25.6. 
29.3 
33.0 
36.6 
73:3 
110.0 

146.6; 

183.3 

180 

18.0 
21.0' 
24. Oi 
27.0! 
30.0! 
60. Oi 
90.0 
120. Oi 
150. Oj 

5 

0.5 ; 
0.6 

0.6 i 
0.7 ' 
0.8 1 

1.6 ! 
2.5 ' 
3.3 



0.4 
0.5 
0.6 
0.7 
0.7 
1.5 
2.2 
3.0 
3 



4 

0.4 
0.4 
0.5 
0.6 
0.6 
1.3 
2.0 
2 _ 

3-3 



3 

0.3 
0.3 
0.4 
0.4 
0.5 
1.0 
1.5 
2.0 
2.5 



2 

0.2 
0.2 
0.2 
0.3 
0.3 
0.6 
1.0 
1.3 
1 



P. P, 





O.Q 
O.Q 
0.0 , 
0.1 
0-1 
0.1 I 
0-2 
3 i 
OjJ 



86t 



TABLE VII.- 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



175* 



- ' Log. Sin. d. Log. Tan. c.d. Log. Cot. Log. Cos. 



78 8 



.84 464 
.84 645 
.84 826 
85 005 
85 184 



.85 863 
.85 540 
.85 717 
.85 893 
■ 86 068 



.87 828 
.87 995 
.88 160 

• 88 326 

• 88 490 



• 88 654 

• 88 817 
.88 980 

• 89 142 
■ 89 303 



.89 464 
.89 624 

■ 89 784 

■ 89 943 
90 lOl 



39 



■ 90 259 

• 90 417 

■ 90 573 

■ 90 729 

• 90 885 



• 91 040 

■ 91 195 

■ 91349 

■ 91 502 

■ 91 655 

■ 91 807 
.91 959 
.92 110 

■ 92 261 

■ 92 411 



■ 92 561 

■ 92 710 

• 92 858 

• 93 007 

• 93 154 



• 93 301 
.93 448 

• 93 594 

• 93 740 

• 93 885 



• 94 029 



Log. Cos, 



^8 



.86 243 
.86 417 
■ 86 590 
.86 763 
• 86 935 



178 
177 
176 
176 
175 



87 953 

88 120 
88 287 
88 453 
88 618 



.88 783 
.88 947 

• 89 111 

• 89 274 

• 89 436 



• 89 598 

■ 89 759 

■ 89 920 

• 90 080 

■ 90 240 



■ 90 398 
.90 557 

■ 90 714 

■ 90 872 
• 91 028 



■91 184 

■ 91 340 

■ 91 495 

■ 91 649 

■ 91 803 



.91 957 
.92 109 
92 262 
.92 413 
• 92 565 



• 92 715 

• 92 866 

• 93 015 

• 93 164 

• 93 313 



• 93 461 

• 93 609 

• 93 756 

• 93 903 

• 94 049 



8^94 195 
Log. Cot. 



.15 535 
.15 354 
.15 174 
.14 994 
.14 815 



.14 637 
.14 459 
.14 283 
.14 107 
.13 931 



.13 756 
.13 582 
.13 409 
.13 237 
.13 065 



156 
155 
155 
154 
154 
153 
152 
152 
15l 
15l 



C.d. 



.12 893 
.12 723 
.12 553 
.12 384 
.12 215 



.12 047 
.11880 
.11 713 
.11 547 
.11 38l 



.11 216 
.11052 
.10 889 
.10 726 
■ 10 563 



.10 401 
.10 240 
.10 079 
.09 91§ 
■ 09 760 



■ 08 815 
.08 660 

• 08 505 
.08 350 

• 08 196 



• 08 043 
.07 890 
.07 738 
.07 586 

• 07 435 



■ 07 284 
.07 134 

■ 06 984 
.06 835 
• 06 686 



■ 06 538 

• 06 390 
.06 243 
.06 097 
J)5 950 

• 05 805 
Log. Tan, 



■ 99 889 

• 99 888 
.99 888 

• 99 887 



• 99 885 
.99 884 
.99 883 
■ 99 882 

• 99 881 



.99 880 
.99 879 
.99 878 
■ 99 877 
• 99 876 



• 99 856 

• 99 855 
.99 853 

• 99 852 

• 99 851 



.99 850 

• 99 849 
■ 99 848 

• 99 847 

• 99 846 



• 99 845 

• 99 844 

• 99 843 

• 99 842 

• 99 841 



• 99 840 

• 99 839 

• 99 837 

• 99 836 

• 99 835 



• 99 834 



94° 



Log. Sin. 
609 



P. P. 





181 


180 


178 


6 


18.1 


18.0 


17.8 


7 


21 


1 


21 





20.7 


8 


24 


1 


24 





23.7 


9 


27 


1 


27 





26^7 


10 


30 


1 


30 





29^6 


20 


60 


3 


60 





59^3 


30 


90 


5 


90 





89.0 


40 


120 


6 


120 





118^6 


50 


150 


8 


150 





148.3 





174 


173 


170 


6 


17^4 


17.2 


17.0 


7 


20 


3 


20.0 


19^8 


8 


23 


2 


22^9 


22^6 


9 


26 


1 


25.8 


25^5 


10 


29 





28^6 


28 • 3 


20 


58 





57.3 


56^6 


30 


87 





86^0 


85^0 


40 


116 





114.6 


113.3 


50 


145 





143.3 


141.6 





16( 


5 


164 


162 


6 


16^6 


16.4 


16^2 


7 


19 


3 


19^1 


18.9 


8 


22 


1 


21.8 


21^6 


9 


24 


9 


24.6 


24^3 


10 


27 


6 


27.3 


27.0 


20 


55 


3 


54-6 


54^0 


30 


83 





82^0 


81^0 


40 


110 


6 


109^3 


108-0 


50 


138 


3 


136.6 


135.0 



176 

17.6 

20.5 
23.4 
26.4 
29.3 
58.6 
880 
117^3 
146.6 

168 

16.8 
19.6 
22.4 
25.2 
28^0 
56-0 
84. 
112.0 
140.0 

160 

16.0 
18^6 
21^3 
24^0 
26.6 
53.3 
80.0 
106 ■ 6 
133-3 





158 


156 


154 


152 


6 


15-8 


15.6 


15^4 


15-2 


7 


18 


4 


18.2 


17 


9 


17 


7 


8 


21 





20^8 


20 


5 


20 


2 


9 


23 


7 


23.4 


23 


1 


22 


8 


10 


26 




26.0 


25 


6 


25 


3 


20 


52 


6 


52.0 


51 


3 


50 


6 


30 


79 





78^0 


77 





76 





40 


105 


3 


104^0 


102 


6 


101 


3 


50 


131 


6 


130^0 


128 


3 


126 


6 





150 


149 


148 


147 


6 


15^0 


14.9 


14.8 


14.7 


7 


17 


5 


17 


4 


17 


2 


17 


1 


8 


20 





19 


8 


19 


7 


19 


6 


9 


22 


5 


22 


3 


22 




22 





10 


25 





24 


8 


24 


6 


24 


5 


20 


50 





49 


g 


49 


3 


49 





30 


75 





74 


5 


74 





73 


5 


40 


100 





99 


3 


98 


6 


98 





50 


125 





124 


1 


123 


3 


122 


5 





146 


145 


T 


1 


6 


14-6 


14^5 


0^1 


0.1 


7 


17^0 


16-9 


0-2 


0. 


8 


19^4 


19^3 


0^2 


0^1 


9 


21^9 


21-7 


0^2 


0^1 


10 


24^3 


24-1 


0.2 


0. . 


20 


48^6 


48^3 


0.5 


0.31 


30 


73.0 


72^5 


0.7 


0.5 


40 


97.3 


96.6 


1.0 


0.6 


50 


121 • 6 


120^8 


1.2 


0.8 



o 

O.Q 
0^0 
0.0 
0.1 
0.1 
0.1 
0.2 
0.3 
0.4 



P. P. 



86^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



174® 



5 

6 

7 

8 

9^ 

10 

11 

12 

13 

U 

15 

16 

17 

18 

19. 

20 

21 
22 
23 
24 



25 

26 

27 

28 

29^ 

30 

31 

32 

33 

3£ 

35 

36 

37 

38 

39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



Log. Sin< 



8.94 029 
8.94 174 
8-94 317 
8.94 46(5 
8.94 603 



8.94 745 

8.94 887 

8.95 028 
8-95 169 
895 310 




8.97 496 
8-97 629 
8-97 762 
8-97 894 
8-98 026 



8-98 157 
8-98 288 
8-98 419 
8-98 549 
8-98 679 



8-98 808 
8-98 937 
8-99 066 
899 194 
8-99 322 



d. 



8-99 449 
8-99 577 
8-99 703 
8-99 830 
8-99 956 



9-00 081 
9-00 207 
9-00 332 
9-00 456 
9-00 585 



9-00 704 
9-00 828 
9-00 951 
9-01 073 
9-01 198 



50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 9-01 923 



9-01 318 
9-01 440 
9-01 561 
9-01 682 
9-01 803 



144 
143 
143 
143 
142 
142 
141 
141 
140 
140 
139 
139 
138 
138 
138 
137 
137 
133 
138 
135 
135 
134 
134 
134 
133 
133 
132 
132 
132 
13l 
131 
130 
130 
130 
129 
129 
128 
123 
127 
127 
127 
126 
128 
126 
125 
125 
125 
124 
124 
124 
123 
123 
122 
122 
122 
122 
12l 
121 
120 
120 



8-97 013 
8-97 149 
8-97 285 
8-97 421 
8-97 556 



8-97 690 
8-97 825 

8.97 958 

8.98 092 
8-98 225 



Log. Tan. c.d. Log. Cot. Log. Cos 



8-94 195 
8.94 340 
8.94 485 
8.94 629 
8.94 773 



8.94 917 

8.95 059 
8.95 202 
8.95 344 
8-95 4851 




8-96 325 
8.96 464 
8-96 602 
8.96 739 
8.96 876 



8-98 357 
8-98 490 
8.98 621 
8.98 753 
8-98 884 



8-99 015 
8.99 145 
8.99 275 
8.99 404 
8-99 533 



8-99 662 
8-99 791 
8-99 919 
9-00 048 
9-00 174 



9-00 30D 
9-00 427 
9-00 553 
9-00 679 
9-00 804 



9-00 930 
9.01 055 
9.01 179 
9.01 303 
9.01 427 



9.01 550 
9.01 673 
9.01 796 
9.01 915 
9-02 040 



9-02 162 



145 
144 
144 
144 
143 
142 
142 
142 
141 
141 
141 
140 
140 
139 
139 
138 
138 
137 
137 
137 
138 
138 
135 
135 
134 
134 
133 
133 
133 
132 
132 
131 
131 
131 
131 
130 
130 
129 
129 
129 
128 
128 
127 
127 
126 
126 
126 
125 
125 

125 

124 
124 
124 
124 
123 
123 
123 
122 
122 
12l 



1.05 805 
1.05 659 
1.05 515 
1.05 370 
1.05 226 



1.05 083 
1.04 940 
1.04 798 
1.04 656 
1.04 514 



1.04 373 
1.04 232 
1.04 092 
1.03 952 
1.03813 



9.99 834 
9.99 833 
9.99 832 
9.99 831 
9.99 830 



9.99 829 
9.99 827 
9.99 826 
9.99 825 
9.99 824 



1.03 674 
1.03 536 
1.03 398 
1.03 260 
1.03 123 



1.02 986 
1.02 850 
1.02 714 
1.02 579 

1.02 444 



1.02 309 
1.02 175 
1.02 041 
1.01 908 
1.01 775 



1.01 642 
1.01 510 
1.01 378 
1.01 247 
1.01 118 



9.99 799 
9.99 798 
9.99 797 
9.99 796 
9.99 794 



1.00 985 
1.00 855 
1.00 725 
1.00 595 
1.00 466 



1.00 337 
1.00 209 
1.00 081 
0.99 955 
0-99 826 



0-99 699 
0-99 573 
0-99 446 
0.99 321 
0-99 195 



0-99 070 
0-98 945 
0-98 821 
0-98 697 
098 573 



0-98 450 
0-98 327 
0-98 204 
0.98 081 
0-97 959 



9.99 823 
9.99 822 
9.99 821 
9.99 81§ 
9.99 8ia 



9.99 81? 
9.99 816 
9.99 815 
9.99 814 
9.99 813 



P.P. 





145 


144 


143 


143 


6 


14.5 


14.4 


14.3 


14.2 


7 


16.9 


16.8 


16.7 


16.5 


8 


19.3 


19.2 


19.0 


18.9 


9 


21.7 


21.6 


21.4 


21.3 


10 


24.1 


24.0 


23.8 


23.6 


20 


48.3 


48.0 


47.6 


47.3 


30 


72.5 


72.0 


71.5 


71.0 


40 


96.6 


96.0 


95.3 


94.fi 


50 


120.8 


120.0 


119.1 


118.3 




9.99 793 
9.99 792 
9.99 791 
9.99 789 
9.99 788 



9.99 787 
9.99 786 
9.99 784 
9.99 783 
9.99 7,82 



9.99 781 
9.99 779 
9.99 778 
9.99 777 
9.99776 



9.99 774 
9.99 773 
9.99 772 
9.99 770 
9.99 769 



9.99 768 
9.99 766 
9-99 765 
9-99 764 
999 763 



0-97 838 9-99 781 





140 


139 


13S 


137 


6 


14-0 


13-9 


13.8 


13.7 


7 


16.3 


16-2 


16.1 


16.0 


8 


18.6 


18.5 


18.4 


18.2 


9 


21.0 


20-6 


20-7 


20-5 


10 


23.3 


23-1 


23-0 


22.8 


20 


46.6 


46-3 


46-0 


45.6 


30 


70.0 


69.5 


69-0 


68.5 


40 


93.3 


92.6 


92-0 


91.§ 


50 


116.6 


115.8 


115-0 


114.i 



141 

14.1 
16.4 
18.8 
21.1 
23.5 
47..0 
70.5 
94.0 
117.5 

136 

13-6 
15.g 
18.1 
20.4 
22-6 
45-3 
68-0 
90-6 
113-3 



9 
10 
20 
30 
40 
60 



6 

7 

8 

9 

10 

20 

30 

40 

50 



135 


134 


133 


13.5 


13.4 


13.3 


15-7 


15.6 


15-5 


18-0 


17-8 


17-7 


20-2 


20-1 


19-9 


22-5 


22-3 


22-1 


45-0 


44-6 


44-3 


67-5 


67-0 


66-5 


90-0 


89-3 


88-6 


112.5 


111.6 


110-8 



131 

13-1 
15.3 
17-4 
19.6 
21-8 
43.6 
65-5 
87.3 
109.1 



130 

13.0 
15.1 
17.3 
19-5 
21-6 
43-3 
65-0 
86-6 
108.3 



139 

12-9 
15-0 
17-2 
19-3 
21-5 
43-0 
64-5 
86-0 
107.5 



133 

13.2 
15.4 
17.6 
19.8 
22.0 
44.0 
68.0 
88.0 
IIO.Q 

138 

12.8 
14.9 
17.0 
19.2 
21.3 
42.6 
64.0 
85.3 
106.6 



137 


136 


135 


134 


133 


12-7 


12-6 


12-5 


12.4 


12-3 


14-8 


14-7 


14-6 


14-4 


14-3 


16-9 


16-8 


16-6 


16-5 


16-4 


19-0 


18-9 


18-7 


18-6 


18-4 


21.1 


21-0 


20.8 


20-6 


20-5 


42-3 


42-0 


41-6 


41-3 


41-0 


63-5 


63-0 


62-5 


62-0 


61-5 


84-6 


84-0 


83-3 


82-6 


820 


105.8 


105-0 


104-1 


103.3 


102-5 





133 


131 


130 


T 


1 


6 


12-2 


12.1 


12-0 


0-1 


0-1 


7 


14.2 


14.1 


14-0 


0-2 


0-1 


8 


16.2 


16.1 


16.0 


0-2 


0-1 


9 


18.3 


18.1 


18.0 


0.2 


0-1 


10 


20.3 


20.1 


20.0 


0-2 


0-1 


20 


40.6 


40.3 


40.0 


0-5 


0-3 


30 


61.0 


60.5 


60.0 


0-7 


0.5 


40 


81.8 


80.6 


80.0 


1-0 


0-6 


50 


101.6 


100.6 


100.0 


1.2 


0.8 



0^ 

n 

0.1 
0.1 
O.I 

0.5 
0.3 

0.4 



Log. Cos, 



Log. Cot.lc.d. Log.Tan. Log. Sin. 



P.P. 



96« 



610 



84^ 



TABLE 



Vn.— LOGARITHMIC SINES, COSINES, TANGJiiJNTS. 
AND COTANGENTS. 



I'J'S'* 



Log. Sin. 



01 923 

02 043 
02 163 
02 282 
02 401 



d. 



Log. Tan, 



02 520 
02 638 
02 756 
02 874 
02 992 



03 109 
03 225 
03 342 
03 458 
03 574 



03 689 
03 805 

03 919 

04 034 
04 148 



04 262 
04 376 
04 489 
04 602 
04 715 



04 828 

04 940 

05 052 
05 163 
05275 



05 386 
05 496 
05 607 
05 717 
05 827 



05 936 

06 046 
06 155 
06 264 
06 37^ 



06 480 
06 588 
06 696 
06 803 
06 910 



07 017 
07 124 
07 230 
07 336 
07 442 



07 548 
07 653 
07 758 
07 863 
07 967 



08 072 
08 176 
08 279 
08 383 
08 486 



9-08 589 



120 
119 
119 
119 
119 
118 
118 
118 
117 
117 
116 
116 
116 
116 
115 
115 
114 
114 
114 
114 
113 
113 
113 
113 
112 
112 
112 
111 
111 
111 
110 
110 
110 
110 
109 
109 
109 
109 
108 
108 
108 
107 
107 
107 
107 
106 
106 
106 
106 
105 
105 
105 
104 
104 
104 
104 
103 
103 
103 
103 



12l 



Log. Cos. 



02 162 ,.^, 

02 283 l^t 

.02 404. {^^ 

,.02 525 |5^ 

9-02 645 ^"^^ 

120 



9.02 765 Its 

9.02 885 ^^^ 

9.03 004 
9.03 123 
9.03 242 



9.03 361 
9.03 479 
9.03 597 
9.03 714 
9.03 83i 



9.03 948 

9.04 065 
9.04 181 
9.04 297 
9.04 413 



d. 



c.d, 



Log. Cot, 



119 
19 



119 
118 



118 



0.96 639 
nnO.96 521 
US 0.96 403 
■^^' 0.96 285 

0.96 168 



9.04 528 
9-04 643 
9.04 758 
9.04 872 
9-04 987 



9.05 101 
9.05 214 
9.05 327 
9.05 440 
9.05 553 



9.05 666 
9.05 778 

9.05 890 

9.06 OOl 
9.06 113 



9.06 224 
9.06 335 
9.06 445 
9.08 555 
9.06 665 



9.06 775 
9.06 884 

9.06 994 

9.07 102 
9-07 2ll 



9.07 319 
9.07 428 
9.07 635 
9.07 643 
9.07 755 



9.07 857 

9.07 962 

9.08 071 
9=08 177 
9.08 283 



9.08 389 
9.08 494 
9.08 600 
9.08 705 
9.08 810 



9-08 914 



Log. Cot, 



Log. Cos, 



0.97 838 
0.97 716 
0.97 595 
0.97 475 
0.97 354 



0.97 234 
0.97 115 
0.96 995 
0.96 876 
0.96 757 



9.99 76T 60 
9.99 760 
9.99 75S 
9.99 757 
9.99 756 



9.99 754 
9.99 753 
9.99 752 
9-99 750 
9-99 749 



117 
117 
116 
116 
116 
115 
115 
115 
114 
114 
114 

114 
113 
113 
113 
113 
112 
112 
112 
111 
111 

111 
111 
110 
110 
liO 
109 
109 
109 
108 
109 
108 
1-08 
107 
107 
107 
107 
107 
106 
106 
106 
105 
105 
105 
105 
105 
104 



0>96 05l 
0-95 935 
0.95 818 
0-95 702 
0.95 587 
0.95 471 
0.95 356 
0.95 242 
0.95 127 
0.95 013 



9-99 748 
9-99 746 
9.99 745 
9-99 744 
9.99 742 



0.94 899 
0.94 785 
0.94 672 
0.94 559 
0.94 446 



9.99 741 
9.99 739 
9.99 738 
9-99 737 
9.99 735 
9-99 734 
9.99 732 
9.99 731 
9-99 730 
9.99 728 
9.99 727 
9.99 725 
9.99 724 
9.99 723 
9.99 721 



0.94 334 
0.94 222 
0.94 110 
0.93 998 
0.93 887 



0.93 665 
0.93 554 
0.93 444 
0.93 334 



0.93 225 
0.93 liB 
0.93 006 
0.92 897 
0-92 788 



0.92 680 
0.92 572 
0.92 464 
0.92 357 
0-92 249 



0.92 142 
0.92 035 
0.91 929 
0.91 822 
0.91.716 



0.91 611 
0.91 505 
0.91 400 
0.91 295 
0.91 190 
0-91 085 
Log. Tan. 



9.99 720 
9.99 718 
9.99 717 
9.99 715 
9.99 714 



0.93 7769.99 712 



9.99 711 
9-99 710 
9.99 708 
9.99 707 



9.99 705 
9.99 704 
9.99 702 
9-99 701 
9.99 699 



9.99 698 
9.99 696 
9.99 695 
9.99 693 
9.99 692 



9.99 690 
9.99 689 
9.99 687 
9.99 686 
9.99 684 



9.99 683 
9.99 681 
9.99 679 
9.99 678 
9.99 676 



P.P. 



9.99 675 
Log. Sin. 

611 





131 


131 


120 


119 


6 


12.1 


12.1 


12.0 


11.9 


7 


14.2 


14.1 


14.0 


13.9 


8 


16.2 


16.1 


16.0 


15.8 


9 


18.2 


18.1 


18.0 


17-8 


10 


20.2 


20.1 


20.0 


19.8 


20 


40.5 


40.3 


40.0 


39.6 


30 


60.7 


60-5 


60.0 


59.5 


40 


81.0 


80.6 


80.0 


79.3 


50 


101.2 


100.8 


100.0 


99.1 



118 

11.8 
13-7 
15.7 
17.7 
19.6 
39.S 
59-0 
78-6 
98.3 



117 

ill.7 
13.7 
I 15-6 
I 17.6 
119.6 
139.1 
3058-7 
4078.3 
5097.9 



6 
7 
8 

9 

10 
20 



117 116 115 



11.7 11.6 
13.6 13-5 
15.6 15.4 
17.5 17.4 
19.5il9.3 
39.038.6 
58.5;58.0!57.5 
78.0,77.3 76.6 
97.596.6 95.8 



11.5 
13-4 
15.3 
17.2 
19.1 
38-3 



6 

7 

8 

9 

10 

20 

30 

40 

50 



115114 113 113 111 

11.3lll.2!ll.i 
13.D 12.9 



11.4 
13.3 
15.2 
17.2 
19.1 
38.1 
57.2 
76.3 
95.4 



11.4 
13.3 
15.2 
17.1 
19-0 
38.0 
57.0 
76.0 
95.0 



13.2 
15.0 
16.9 
18.8 



14.9 14.8 
16.8 16.6 
18-6 18.5 
37.6!37.3,37.0 
56.5l56.0;55.5 
75.3,74.6 74.0 
94.1193.3192.5 



6 
7 

8 
9 
10 
20 
30 
40 
50 



110 110 

ll.Olll.O 
12.9 12.8 
14.7114.6 
16.616.5 
I8.4II8.3 
36.8:36.6 
5o.2 50.0 
73.673-3 
92.ll9x.6 



109 108 

io.9;io.8 

12.7|12.6 
14.5 14.4 
16.3 16o2 
18.1 180 
3b. 3 36-0 
5^.5 5y:.0 
7:^.6 72.0 
9u.8 9u.O 



6 
7 

8 
9 
10 
20 
30 
40 
50 



107 

10.7 
12.5 
14.3 
16.1 
17-9 
35-8 
53-7 
71-6 
89.6 



107 

10.7 
12.5 
14-2 
16.0 
17.8 
35.6 
53.5 
71.3 
89.1 



106 105 104 



10.6 
12.3 
14.1 
15.9 
17.6 
35.3 
53.0 
70.6 
88.3 



6 
7 

8 
9 
10 
20 
SO 
40 
_50 



105 

10.3 
12.1 
13-8 
15.5 
17.2 
34-5 
51.7 
69-0 
86.2 



103 

10.3 
12.0 
13.7 
15.4 
17-1 
34.3 
51.5 
68.6 
85-8 



10.5 
12.2 
14.0 
15.7 
17.5 
35.0 
52.5 
70.0 
87.5 

T 
O.I 
0.2 

0-2 
0.2 
0.2 
0.5 
0.7 
1.0 
1.2 



10.4 
12.1 
13.8 
15.6 
17^ 
34.6 
520 
69.3 
86.6 

1 

0.1 
0.1 
0.1 
O.I 
O.I 

0.3 
0.5 

0.6 
0.8 



P.P. 



83* 



TABLE VII.- 



-LOGARITHMIC SINES, COSINES. TANGENTS, 
AND COTANGENTS. 



172F 



' Log. Sin. d. Log. Tan. cd. Log. Cot. Log. Cos 



08 589 
08 692 
08 794 
08 897 
08 999 



09 101 
09 202 
09 303 
09 404 
09 505 



09 606 
09 706 
09 806 

09 906 

10 006 



10 105 
10 205 
10 303 
10 402 
10 501 



10 599 
10 697 
10 795 
10 892 
10 990 



11087 
11 184 
11281 
11377 
11473 



11570 
11 665 
11761 
11 856 
11 952 



12 047 
12 141 
12 236 
12 330 
12425 



12 518 
12 612 
12 706 
12 799 
12 892 



12 985 

13 078 
13 170 
13 263 
13 355 



13 447 
13 538 
13 630 
13 721 
13 813 



13 903 

13 994 

14 085 
14 175 
14 265 



14 355 



Log. Cos 



102 
102 
102 
102 
102 
101 
101 
101 
101 
100 
100 
100 
100 
99 
99 



10 454 
10 555 
10 655 
10 756 
10 858 



10 956 
11055 
11155 
11254 
11353 



11452 
11550 
11 649 
11 747 
11845 



08 914 

09 018 
09 123 
09 226 
09 330 



09 433 
09 536 
09 639 
09 742 
Oa844 



09 947 

10 048 
10 150 
10 252 
10 353 



11943 
12 040 
12 137 
12 235 
12 331 



104 
104 
103 
103 

103 

103 

103 

102 

102 

102 

101 

102 

101 

101 

101 

101 

100 

100 

100 

100 

99 

99 

99 

99 

98 

98 

98 

98 

98 

98 

97 

97 

97 



12 428 
12 525 
12 621 
12 717 
12 813 



12 908 

13 004 
13 099 
13 194 
13 289 



13 384 
13 478 
13 572 
13 666 
13 760 



13 854 

13 947 

14 041 
14 134 
14 227 



14 319 
14 412 
14 504 
14 596 
14 688 



14 780 



91085 
90 981 
90 877 
90 773 
90 670 



90 566 
90 463 
90 360 
90 258 
90 15 5 



9-99 
9.99 



9.99 
9.99 
9.99 
9.99 
9.99 



675 
673 
672 
670 
669 
667 
665 
664 
662 
661 



0. 



0. 



0. 



90 053 
89 951 
89 849 
89 748 

89 647 



89 546 
89 445 
89 344 
89 244 
89 144 



89 044 
88 944 
88 845 
88 745 
88 646 



9.99 
9.99 
9.99 
9.99 
9.99 



88 548 
88 449 
88 351 
88 253 
88 155 



9.99 
9.99 
9.99 
9.99 
9.99 



88 057 
87 959 
87 862 
87 765 
87 668 



87 571 
87 475 
87 379 
87 283 
87 187 



87 091 
86 996 
86 900 
86 805 
86 710 



86 616 
86 521 
86 427 
86 333 
86 239 



86 146 
86 052 
85 959 
85 866 
85 773 



85 680 
85 588 
85 495 
85 403 
85 311 



.85 219 



9.99 
99 
9.99 
9.99 
9.99 



659 
658 
656 
654 
653 



9.99 
9.99 
9.99 
9.99 
9.99 



651 
650 
648 
646 
645 



643 
641 
640 
638 
637 
635 
633 
632 
630 
628 



9.99 
9.99 
9.99 
9.99 
9.99 

9.99 
9.99 
9.99 
99 
9.99 



627 
625 
623 
622 
620 



618 
617 
615 
613 
611 



9.99 
9.99 
9.99 
9-99 



610 
608 
606 
605 
603 



9.99 
9.99 
9.99 
9.99 
9-99 



601 
600 
598 
596 
594 



9.99 
9.99 



9.99 
9.99 
9.99 
9. 99 
9^91 
9. 99 



9T 



d. Log. Cot.[c.d. Log, Tan. Log. 
6I2" 



593 
591 
589 
587 
586 
584 
582 
580 
579 
577 
575 
Sin 



P. P. 





104 


103 


6 


10.4 


10.3 


7 


12.1 


12.0 


8 


13.8 


13.7 


9 


15.6 


15.4 


10 


17.3 


17.1 


20 


34.6 


34.3 


30 


52.0 


51.5 


40 


69.3 


68.6 


50 


86.6 


85.8 



103 

10.2 
11.9 
13.6 
15.3 

17. 

34.0 
51.0 
68.0 
85.0 



101 

10-1 
11.8 
13.4 
15.1 
16-8 
33.6 
50.5 
67.1 



84 





100 


100 


99 


6 


10.0 


10.0 


9.9 


7 


11.7 


11.6 


11.5 


8 


13.4 


13.3 


13.2 


9 


15.1 


15.0 


14.8 


10 


16.7 


16.6 


16.5 


20 


33.5 


33.3 


33.0 


30 


50.2 


50.0 


49.5 


^0 


67.0 


66.6 


66.0 


50 


83.7 


83. S 


82.5 





97 


97 


96 


6 


9.7 


9.7 


9.6 


7 


11. 4 


11.3 


11.2 


8 


13.0 


12.9 


12.8 


9 


14.6 


14.5 


14.4 


10 


16.2 


16.1 


16.0 


20 


32.5 


32.3 


32 a) 


30 


48.7 


48.5 


48.0 


40 


65.0 


64.6 


64.0 


50 


81.2 


80.8 


80.0 



6 

7 

8 

9 

10 

20 

30 

40 

50- 



92 

9.4 

11.0 
12.6 
14.2 
15.7 
31.5 
47.2 
63.0 
78.7 



94 

9.4 
10.9 
12.5 
14.1 
15.6 
31 3 
47.0 
62.6 
78-3 



93 

9.3 
10.8 

12.4 
13.9 
15.5 
31.0 
46.5 
62.0 
77.5 



98 

9.8 

11.4 
13.0 
14.7 
16. 3 
32.6 
49.0 
65. § 
81.6 



95 

9.5 
11.1 
12.6 
14.2 
15.8 
31.6 
47.5 
63.3 
79.1 



92 

9.2 
10.? 
12.2 
13.8 
15.3 
30.6 
46. Q 
61. 1 
76.6 





9.1 


91 


90 


2 


6 


9.1 


9.0 


0.2 


7 


10.7 


10 


] 


10.5 


0.2 


8 


12.2 


12 




12.0 


0.2 


9 


13-7 


13 




13.5 


0.3 


10 


15.2 


15 




15.0 


0.3 


20 


30 . 5 


30 


c 


30.0 


0.6 


30 


45.7 


45 


5 


45.0 


1.0 


40 


61.0 


60 


6 


60.0 


1 -3 


50 


76.2 


75 


8 


75.0 


1-6 



oh 

0.2' 
0.2 
0.2 
0.2 
0.5 
0.7 
1.0. 
1.2- 



P. P. 



8?8^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



171' 



"* •' Log. Sin. d. Log. Tan. c.d. Log. Cot. Log. Cos 



9.15 
9.15 
9-15 
9.15 
9.15 



9.15 
9.15 
9.15 
9.15 
9-16 



9.16 
9.16 
9.16 
9.16 
9.16 



9.14 
9.14 
9.14 
9.14 
9.14 



9.14 
9.14 
9.14 
9.15 
9.15 



9.16 
9.16 
9.16 
9.16 
9.16 
9.16 
9.17 
9.17 
9.17 
9.17 



9.17 
9.17 
9.17 
9-17 
9-17 



9.17 
9-17 
9.17 
9.18 
9.18 



9.18 
9.18 
9-18 
9.18 
9.18 



9.18 
9.18 
918 
9.18 
918 
9.19 
9.19 
9- 19 
9-19 
9.19 
9.19 



355 
445 
535 
624 
713 
802 
891 
980 
068 
157 
245 
333 
421 
508 
595 
683 
770 
857 
943 
030 
116 
202 
288 
374 
460 

545 
630 
716 
801 
885 
970 
054 
139 
223 
307 
391 
474 
558 
641 
724 
807 
890 
972 
055 
137 
219 
301 
383 
465 
546 
628 
709 
790 
871 
952 
032 
113 
193 
273 
353 
433 



Log. Cos. 



9.16 577 
9.16 665 
9.16 753 
9.16 841 
9.16 928 



9.17 015 
9.17 103 
9.17 190 
9.17 276 
17 363 



9.14 780 

9.14 872 
9-14 963 

9.15 054 
15 145 



9.15 236 
15 327 
9.15 417 
9.15 507 
9.15 598 




17 450 

17 536 

9-17 622 

9.17 708 

9.17 794 



9.17 800 

9.17 965 
18 051 

9.18 136 
18 221 



• 18 306 

• 18 390 

• 18 475 
.18 559 

18 644 




9.19 560 

19 643 

19 725 

9.19 807 

9.19 889 



9.19 971 
Log. Cot, 



85 219 
85 128 
85 037 
84 945 
84 854 



84 763 
84 673 
84 582 
84 492 
84402 



84 312 
84 222 
84 133 
84 043 
83 954 



83 865 
83 776 
83 687 
83 599 
83 511 



0.83 422 
0.83 334 
0.83 247 
0.83 159 
0. 83 071 



0.82 984 
0.82 897 
0.82 810 
0.82 723 
0.82 636 



0.82 550 
0.82 464 
0.82 377 
0.82 291 
0.82 206 



0.82 120 
0-82 034 
0.81 949 
0.81 864 
0-81 779 



0.81 694 
0.81 609 
0-81 525 
0.81440 
0.81 356 



0.81 272 
0.81 188 
0.81 104 
0.81 020 
0-80 937 



0.80 854 
0.80 770 
0.80 687 
0-80 604 
0.80 522 



0.80 439 
0. 80 357 
0.80 274 
0.80 192 
80 XIO 



Q. 80 028 
Log, Tan. 




99 575 
99 573 
99 571 
99 570 
99 568 



99 548 
99 546 
99 544 
99 542 
99 541 



539 
537 
535 
533 
531 



99 



529 
528 
526 
524 
522 



520 
518 
516 
514 
512 



511 
509 
507 
505 
503 
501 
499 
497 
495 
493 



491 
489 
487 
485 
484 



99 



9. 99 



482 
480 
478 
476 
474 
472 
470 
468 
466 
464 
462 



Log. Sin, 
613 



P. P. 





91 


91 


90 


6 


9.1 


9.1 


9-0 


7 


10.7 


10.6 


10 


5 


8 


12.2 


12.1 


12 





9 


13-7 


13-6 


13 


5 


10 


15.2 


15.1 


15 





20 


30.5 


30.3 


30 





30 


45.7 


45.5 


45 





40 


61.0 


60.6 


60 





50 


76.2 


75-8 


75 








88 


88 


87 


86 


6 


8 8 


88 


8-7 


8. 


7 


10.3 


10 


2 


10-1 


10. 


8 


11.8 


11 


7 


11.6 


11. 


9 


13.3 


13 


2 


13.0 


12. 


10 


14-7 


14 


5 


14.5 


14. 


20 


29-5 


29 


3 


29.0 


28. 


30 


44-2 


44 





43.5 


43. 


40 


59-0 


58 




58.0 


57. 


50 


73.7 


73 


3 


72.5 


71. 





85 


85 


84 


6 


8.5 


8.5 


8.4 


7 


10.0 


9 


9 


9.8 


8 


11.4 


11 


3 


11.2 


9 


12.8 


12 




12.6 


10 


14.2 


14 


' 


14.0 


20 


28.5 


28 


3 


28.0 


30 


42 = 7 


42 


5 


42.0 


40 


57.0 


56 


6 


56.0 


50 


71.2 


70 


8 


70.0 



83 
8-2 



83 

8.2 

9 

10 

12 

13 

27.3 
41-0 
54.6 
68-3 



81 

1 

4 



89 
8.9 
10.4 
11.8 
13-3 
14.8 
29.6 
44.5 
59.3 
74.1 



83 
8.3 

9.7 
11.0 
12.5 
13.8 
27.6 
41.5 
55.3 
69.1 



80 

80 





79 


2 


i 


6 


7.9 


0-2 


0- 


7 


9 


3 


02 


0. 


8 


10 


6 


0.2 


0- 


9 


11 


9 


0.3 


0. 


10 


13 


2 


0.3 


0. 


20 


26 


5 


0-6 


0. 


30 


39 


7 


1-0 


0. 


40 


53 





1.3 


1. 


50 


66 


2 


1.6 


1. 



p.p. 



81^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 



170P 



Log. Sin 



19 433 
19 513 
19 592 
19 672 
19 75] 



19 830 
19 909 

19 988 

20 066 
20 145 



20 223 
20 301 
20 379 
20 457 
20 535 



20 613 
20 690 
20 768 
20 845 
20 922 



20 999 

21 076 
21 152 
21 229 
21 305 



21382 
21 458 
21 534 
21 609 
21 685 



21 761 
21836 
21 9ll 

21 987 

22 062 



22 136 
22 211 
22 288 
22 360 
22 435 



22 509 
22 583 
22 657 
22 731 
22 805 



22 878 

22 952 

23 025 
23 098 
23 171 



23 244 
23 317 
23 390 
23 462 
23 535 



23 607 
23 679 
23 751 
23 823 
23 895 



9-23 967 
Log. Cos 



9.21 578 
9.21 657 
9.21 735 
9.21814 
9.21 892 



9.21 971 
9 . 22 049 

9.22 127 
9.22 205 
9.22 283 



Log. Tan 



9.19 971 

9.20 053 
9.20 134 
9.20 216 
9.20 297 



9.20 378 
9.20 459 
9.20 540 
9.20 620 
9.20 701 



9.20 781 

9.20 862 
9-20 942 

9.21 022 
9.21 102 



9.21 181 
9.21 261 
9.21 340 
21 420 
9.21 499 



9.22 360 
9.22 438 
9.22 515 
9.22 593 
9-22 670 



9.22 747 
9-22 824 
9.22 900 
9.22 977 
9-23 054 



9.23 130 
9.23 206 
9.23 282 
9 23 358 
9.23 434 



9.23 510 
9.23 586 
9.23 661 
9.23 737 
9.23 812 



9.23 887 

9.23 962 

9.24 037 
9.24 112 
9.24 186 



9.24 261 
9.24 335 
9.24 409 
9 . 24 484 
9.24 558 
9 ^24 632 
Log. Cot 



c.d 



0.78 
0.78 
0.78 
0.78 
78 



0.78 422 
0.78 343 
0.78 264 
0.78 186 
0.78 107 



Log. Cot. Log. Cos. 



80 028 

79 947 

0.79 865 

0-79 784 

0.79 703 



0.79 
0.79 
0.79 
0-79 
0.79 



622 
541 
460 
379 
298 



0.79 
0.79 
0.79 
0.78 
0.78 



218 
138 
058 
978 
898 
818 
739 
659 
580 
501 



0.78 
0.77 
0.77 
0.77 
0.77 
0.77 
0.77 
0.77 
0.77 
0.77 



029 
951 
873 
795 
717 



562 
484 
407 
330 



0-77 
0.77 
0.77 
0.77 
0.76 



253 
176 
099 
022 
946 



0.76 
0-76 
0.76 
0.76 
0.76 



870 
793 
717 
641 
565 



0.76 
0.76 
0.76 
0.76 
0.76 



489 
414 
338 
263 
188 



0.76 113 
0.76 038 
0-75 963 
075 888 
0.75 813 



0.75 739 
0.75 664 
0.75 590 
0-75 516 
0-75442 



0-75 368 
Log. Tan 




99 442 
99 440 
99 437 
99 435 
99 433 
99 431 
99 429 
99 427 
99 425 
99 423 



99 421 
99 419 
99 417 
99 415 
99 413 



99 411 
99 408 
99 406 
99 404 
99 402 



99 400 
919 398 
99 396 
99 394 
99 392 



99 389 
99 387 
99 385 
99 383 
99 381 



99 379 
99 377 
99 '374 
99 372 
99 370 



99 368 
99 366 
99 364 
99 361 
99 359 



99 357 
99 355 
99 353 
99 350 
99 348 



99 346 
99 344 
99 342 
99 339 
99 337 



999 335 



99* 



Log. Sin 



p. p. 



















81 81 80 79 


6 


8.1 


8.1 


8-0 


7.9 


7 


9 


5 


9 


4 


9 


3 


9.2 


8 


10 


8 


10 


8 


10 


6 


10.5 


9 


12 


2 


12 


1 


12 





11.8 


10 


13 


6 


13 


5 


13 


3 


13.1 


20 


27 


1 


27 





26 


fi 


26.3 


30 


40 




40 


5 


40 





39.5 


40 


54 


3 


54 





53 


3 


52.6 


50 


67 


9 


67 


5 


66 


6 


65.8 



78 


78 


7-8 


78 


9 


1 


9.1 


10 


4 


10.4 


11 


8 


11.7 


13 


1 


13-0 


26 


1 


26-0 


39 


2 


39-0 


52 


3 


52.0 


65 


4 


65.0 



77 
7.7 
9.0 
10.2 
11.5 
12.8 
25.6 
38.5 
51.3 
64. i 





76 


76 


75 


6 


7.6 


7.6 


7.5 


7 


8 


9 


8.8 


8.7 


8 


10 


2 


10-1 


10.0 


9 


11 


5 


11-4 


11.2 


10 


12 


7 


12-6 


12.5 


20 


25 


5 


25.3 


25.0 


30 


38 


2 


38.0 


37.5 


40 


51 





50.6 


50.0 


50 


63 


7 


63.3 


62.5 



74 

li 

9.8 

11.1 
12.3 
24.6 
37.0 
49.3 
61. § 



7 
8 

9 
10 
20 
30 
40 
50 



73 


73 


72 


7.3 


73 


7.2 


8 


6 


8 


5 


8 


4 


9 


8 


9 


7 


9 


6 


11 





10 


9 


10 


8 


12 


2 


12 


1 


12 





24 


5 


24 


3 


24 





36 


7 


36 


5 


36 





49 





48 6 


48 





61 


2 


60 


8 


60 






6 

7| 
8 
9 
10 
20 
30 
40 
50 



71 

7.1 
3 
5 
7 
9 



71 

7.1 
3 

4 
6 



2 

0.2 



P P. 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



169* 



Log. Sin. 



9 



23 967 

24 038 
24 110 
24 181 

24 252 



24 323 
24 394 
24 465 
24 536 
24 607 



24 677 
24 748 
24 818 
24 888 
24 958 



25 028 
25 098 
25 167 
25 237 
25 306 



376 
445 
514 
583 
652 

721 
790, 
858 
927 
995 
063 
131 
199 
267 
335 
402 
470 
537 
605 
672 
739 
806 
873 
940 
007 
073 
140 
206 
272 



405 
471 
536 
602 
668 
733 
799 
864 
929 
995 
60 9-2 8 060 
Cos 



27^ 
27^ 
27 ! 
27 ( 
27 I 
27: 
27 : 
27 J 
27 1 
27 1 



Log. 



26 443 
9. 26 514 
9.26 584 
9.26 655 
9.26 726 



Log. Tan. c. d. Log. Cot. Log. Cos 



24 632 
9.24 705 
9.24 779 
9.24 853 
9-24 926 



9. 25 000 
9.25 073 
9.25 146 
25 219 
9.25 292 



9.25 365 
9.25 437 
9.25 510 
25 582 
9.25 654 



25 727 
25 799 
25 871 

25 943 

26 014 



9.26 086 
9.26 158 
9.26 229 
9.26 300 
9.26 371 



9.26 796 
9.26 867 

9.26 937 

9.27 007 
9.27 078 



9.27 148 
9.27 218 
9.27 287 
9.27 357 
9.27 427 



9.27 496 
9.27 566 
9.27 635 
9.27 704 
9.27 773 



9.27 842 
9.27 911 

9.27 980 

9.28 049 
9.28 117 



9.28 186 
9.28 254 
9.28 322 
9.28 390 
9.28 459 



9.28 527 
9.28 594 
9.28 662 
9-28 730 
9.28 797 



9.28 865 
Log. Cot, 



0.73 557 
0.73 486 
0.73 415 
0.73 344 
0.73 274 



0.75 368 
0.75 294 
0.75 220 
0.75 147 
0-75 073 



0.75 000 
0.74 927 
0.74 854 
0.74 781 
0.74 708 



0.74 635 
0.74 562 
0-74 490 
0. 74 417 
0.74 345 



0.74 273 
0.74 201 
0.74 129 
0.74 057 
0.73 985 



0.73 913 
0-73 842 
0.73 771 
0.73 699 
0.73 628 



0.73 203 
073 133 
0.73 062 
0-72 992 
0.72 922 



0.72 852 
0.72 782 
0.72 712 
0.72 642 
0.72 573 



0.72 503 
0.72 434 
0.72 365 
0.72 295 
0.72 226 



0.72 157 
0.72 088 
0.72 020 
0.71951 
0.71 882 



0.71814 
0.71 746 
0.71 677 
0.71 609 
0.71 541 



0.71 473 
0.71405 
0.71337 
0.71 270 
0.71 202 
0.71 135 
Log. Tan 



99 335 
99 333 
99 330 
99 328 
99326 



99 324 
99 321 
99 319 
99 317 
99 315 



99 312 
99 310 
99 308 
99 306 

99 3or 



99 301 
99 299 
99 296 
99 294 
99 292 



99 290 
99 287 
99 285 
99 283 
99280 



99 278 
99 276 
99 273 
99 271 
99^69 
99 266 
99 264 
99 262 
99 259 
99 257 



99 255 
99 252 
99 250 
99 248 
99 245 



243 
240 
238 
236 
233 
231 
228 
226 
224 
221 



99 



219 
216 
214 
212 
209 
207 
204 
202 
199 
197 



9 

9.99 194 

Log. Sin 



60 

59 
58 
57 
1^ 
55 
54 
53 
52 
II 
50 
49 
48 
47 
_46 

45 
44 
43 
42 
41 
40 
39 
38 
37 
Jl 
35 
34 
33 
32 
II 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16_ 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6_ 
5 
4 
3 
2 
1 



P.P. 



6 
7 
8 

9 

10 
20 
30 
40 
50 



74 

7.4 
8.6 
9.8 
11-1 
12.3 
24.6 
37.0 
49-3 
61.6 



73 

7.3 



73 

7.3 





73 


72 


71 


6 


7.2 


7.2 


7.11 


7 


84 


8.4 


8 


3 


8 


9.6 


9.6 


9 


5 


9 


10.9 


10.8 


10 


7 


10 12.1 


12.0 


11 


9 


20 24.1 


24.0 


23 


3 


30 36.2 


36 


35 


7 


40 48.3 


48.0 


47 


6 


50 60.4 


60.0 


59 


6 





70 


70 


69 


6f 


6 


7.0 


7.0 


6-9 


6. 


7 


8.2 


8 


1 


8 


1 


8. 


8 


9.4 


9 


3 


9 


2 


9. 


9 


10.6 


10 


5 


10 


4 


10. 


10 


11.7 


11 


6 


11 


(^ 


11. 


20 


23.5 


23 


3 


23 


I 


23. 


30 


35.2 


35 





34 


7 


34- 


40 


47.0 


46 


6 


46 


3 


46- 


50 


58.7 


58 


3 


57 


9 


57. 



68 



6 


6 


8 


6 


7 


8 





7 


8 


9 


1 


9 


9 


10 


3 


10 


10 


11 


4 


11 


20 


22 


3 


22 


30 


34 


2 


34- 


40 


45 


6 


45 


50 


57 


1 


56. 


66 6( 


6 


6^6 


6. 


7 


7 


7 


7. 


8 


8 


3 


8. 


9 


10 





9 


10 


11 


1 


11. 


20 


22 


1 


22. 


30 


33 


2 


33 


40 


44 


3 


44 


50 


55 


4 


55. 



68 
8 

• 9 

• 
•2 
•3 
-6 
•Q 
•3 
.6 



67 

6-7 



•5 
7 
•9 
.1 
.3 
•5 
•6 
• 8 

71 

7.1 

8.3 
9.5 
10.6 
11.8 
23.6 
35.5 
47.3 
59.1 



67 

6.7 
8 

i 

I 

3 
5 
6 
8 



65 

"6.5 

7 

8 

9 
10 
21 
32 
43 
54 



65 

6.5 
6 
6 
7 

10.8 
21.6 
32.5 
43 
54 





2 


6 


0.2 


7 


0.3 


8 


0.3 


9 


0.4 


10 


0.4 


20 


0.8 


30 


1 2 


40 


1.6 


50 


2.1 



2 

0.2 
0.2 
0.2 
0.3 
0.3 
0.6 
1.0 
1.3 
1.6 



P.P. 



615 



79* 



TABLE VII.- 



ir 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



168** 



Log. Sin 



28 060 
28 125 
28 189 
28 254 
28 319 



28 383 
28 448 
28 512 
28 576 
28 641 



28 705 
28 769 
28 832 
28 896 
28 960 



29 023 
29 087 
29 150 
29 213 
29 277 



29 340 
29 403 
29 466 
29 528 
29 591 



29 654 
29 716 
29 779 
29 841 
29 903 



29 965 

30 027 
30 089 
30 151 
30 213 



30 275 
30 336 
30 398 
30 459 
30 520 



30 582 
30 643 
30 704 
30 765 
30 826 



30 886 

30 947 

31 008 
31 068 
31 129 



31 189 
31 249 
31 309 
31 370 
31 429 



31489 
31 549 
31 609 
31 669 
31 728 



31 788 



Log. Cos 



Log. Tan. c. d. Log. Cot. Log. Cos 




28 865 

28 932 

29 000 
29 067 
29134 



29 201 
29 268 
29 335 
29 401 
29 468 



30 522 
30 587 
30 652 
30 717 
30 781 



30 846 
30 911 

30 975 

31 040 
31 104 



31 168 
31 232 
31 297 
31 361 
31 424 



31 488 
31 552 
31 616 
31 679 
31743 



31 806 
31 869 
31 933 

31 996 

32 059 



32 122 
32 ■'85 
32 248 
32 310 
32 373 



32 436 
32 498 
32 560 
32 623 
32 685 



9 32 747 
Log. Cot, 



66 
65 

65 
65 
65 
65 
65 

65 
65 
65 
65 
64 
65 
64 
64 
64 
64 
64 
64 
64 
64 
63 
64 
64 
63 
65 
63 
63 
63 
63 
63 
63 
63 
63 
63 
62 
63 
62 
62 
62 
62 
62 
62 



71 135 
71067 
71 000 
70 933 
70 866 



70 798 
70 732 
70 665 
70 598 
70 531 



70 465 
70 398 
70 332 
70 266 
70 200 



70 134 
70 068 
70 002 
69 936 
69 870 



69 805 
69 739 
69 674 
69 608 
69 543 9 



69 478 
69 413 
69 348 
69 283 
69 218 



69 153 
69 089 
69 024 
68 980 
68 896 



68 831 
68 787 
68 703 
68 639 
68 575 



68 511 
68 447 
68 384 
68 320 
68 257 



68 193 
68 130 
68 067 
68 004 
67 941 



67 878 
67 815 
67 752 
67 689 
67 626 



67 564 
67 501 
67 439 
67 377 
67 314 



0-67 252 
Log. Tan 



99 194 
99 192 
99 189 
99 187 
99 185 



99 182 
99 180 
99 177 
99 175 
99 172 



99 170 
99 167 
99 165 
99 162 
99 160 



99 157 
99 155 
99 152 
99 150 
99 147 



99 145 
99 142 
99 139 
99 137 
99 134 



99 132 
99 129 
99 127 
99 124 
99 122 



99 119 
99 116 
99 114 
99 111 
99 109 



99 106 
99 104 
99 101 
99 098 
99 096 



99 093" 
99 091 
99 088 
99 085 
99 083 



99 080 
99 077 
99 075 
99 072 
99 089 



99 087 
99 084 
99 062 
99 059 
99 056 



99 054 
99 051 
99 048 
99 046 
99 043 



99 040 
Log. Sin 



55 
54 
53 
52 

50 

49 
48 
47 

li 

45 
44 
43 
42 
41 

40 

39 
38 
37 
J6^ 

35 
34 
33 
32 
11 
30 
29 
28 
27 
26_ 

25 
24 
23 
22 
21^ 
30 
19 
18 
17 
16_ 
15 
14 
13 
12 
U_ 
10 
9 
8 
7 

_6_ 
5 
4 
3 
2 
1 



P. P. 



67_ 



6 


7 


6. 


7 


9 


7. 


9 





8. 


10 


1 


10. 


11 


2 


11. 


22 


5 


22. 


33 


7 


33- 


45 





44. 


56 


2 


55. 



67 
7 





I 

3 
5 





66 


6( 


8 


6. 


m 
> 


6.* 


6 


6.8 


6.6 


6.5 


6. 


7 


7 


7 


7 


7 


7 


6 


7. 


8 


8 


3 


8 


8 


8 


7 


8. 


8 


10 





9 


9 


9 


8 


9 


10 


11 


1 


11 





10 


g 


10. 


20 


22 


1 


22 





21 


3 


21. 


30 


33 


2 


33 





32 


7 


32. 


40 


44 


3 


44 





43 


6 


43. 


50 


55 


4 


55 





54 


6 


54. 





64 


64 


63 


6r 


6 


6.4 


6.4 


6.3 


6 


7 


7 


5 


7 


4 


7 


4 


7 


8 


8 


6 


8 


5 


8 


4 


8. 


9 


9 


7 


9 


6 


9 


5 


9. 


10 


10 


7 


10 


6 


10 


6 


10. 


20 


21 


5 


21 


3 


21 


1 


21. 


30 


32 


2 


32 





31 


7 


31. 


40 


43 





42 


6 


42 


3 


42. 


50 


53 


7 


53 


3 


52 


9 


52. 





6^ 


63 


61 


6] 


6 


6.2 


6-2 


6.1 


6 


7 


7 


3 


7 


2 


7 


2 


7 


8 


8 


3 


8 


2 


8 


2 


8. 


9 


9 


4 


9 


3 


9 


2 


9. 


10 


10 


4 


10 


3 


10 


2 


10. 


20 


20 


8 


20 


6 


20 


5 


20. 


30 


31 


2 


31 





30 


7 


30. 


40 


41 


6 


41 


3 


41 





40. 


50 


52 


1 


51 


6 


51 


2 


50. 



60 

6.0 

7-0 

8.0 

9.1 

10.1 

20.1 

30.2 

40.3 



60 

6.0 



50.4150 



59 

5.9 

6 

7 

8 

9 
19 
29 
39 
49 





3 


2 , 


6 


0.3 


0-2 


7 


0.3 





3 


8 


0.4 





3 


9 


0.4 





4 


10 


0.5 





4 


20 


1.0 





8 


30 


1.5 


1 


2 


40 


2.0 


1 


6 


50 


2.5 


2 


1 



2 

0.2 
0.2 
0.2 
0.3 
0.3 
0.6 
1.0 
1.3 
1.6 



P. P. 



101° 



616 



78° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



167° 



Log. Sin. d. Log. Tan. c.d. Log. Cot. Log. Cos, 



31 788 
31 847 
31 906 

31 966 

32 025 



32 084 
32 143 
32 202 
32 260 
32 319 



32 378 
32 436 
32 495 
32 553 
32 611 



32 670 
32 728 
32 786 
32 844 
32 902 



32 960 

33 017 
33 075 
33 133 
33 190 



33 248 
33 305 
33 362 
33 419 
33 476 
33 533 
33 590 
33 647 
33 704 
33 761 



33 817 
33 874 
33 930 

33 987 

34 043 



34 099 
34 156 
34 212 
34 268 
34 324 



34 379 
34 435 
34 491 
34 547 
34 602 



34 658 
34 713 
34 768 
34 824 
34 879 



34 934 

34 989 

35 044 
35 099 
35 154 



9 35 209 
Log. Cos 



747 
809 
871 
933 
995 
057 
118 
180 
242 
303 
364 
426 
487 
548 
609 
670 
731 
792 
852 
913 
974 
034 
095 
155 
215 
275 
336 
396 
456 
515 

575 
635 
695 
754 
814 
873 
933 
992 
051 
110 
169 
228 
287 
346 
405 
464 
522 
581 
640 
698 
756 
815 
873 
931 



36 
Log. 



047 
105 
163 
221 
278 
336 
Cot, 




67 252 
67 190 
67 128 
67 066 
67 004 



66 943 
66 881 
66 819 
66 758 
66 696 



66 635 
66 574 
66 513 
66 452 
66 390 



66 330 
66 269 
66 208 
66 147 
66 086 



65 424 
65 364 
65 305 
65 245 
65 186 



65 126 
65 067 
65 008 
64 948 
64 889 



64 830 
64 771 
64 712 
64 653 
64 594 



64 536 
64 477 
64 418 
64 360 
64 302 



243 
185 
127 
068 
010 
952 
894 
837 
779 
721 



0.63 663 
Log. Tan. 



99 040 
99 038 
99 035 
99 032 
99 029 



99 027 
99 024 
99 021 
99 019 
99 016 



99 013 
99 010 
99 008 
99 005 
99 002 



98 999 
98 997 
98 994 
98 991 
98 988 



98 983 
98 980 
98 977 
98 975 



98 972 
98 969 
98 966 
98 963 
98 961 



98 958 
98 955 
98 952 
98 949 
98 947 



98 944 
98 941 
98 938 
98 935 
98 933 



98 930 
98 927 
98 924 
98 921 
98 918 



98 915 
98 913 
98 910 
98 907 
98 904 



98 901 
98 898 
98 895 
98 892 
98 890 



98 887 
98 884 
98 881 
98 878 
98 875 



9. 98 872 

Log. Sin, 



P. P. 



62 61 



6 2 


6.1 


6. 


7 


2 


7-2 


7. 


8 


2 


8.2 


8 


9 


3 


9-2 


9 


10 


3 


10.2 


10 


20 


6 


20-5 


20 


31 





30 7 


30 


41 


3 


41.0 


40 


51 


6 


51 2 


50 



61 

1 
1 
I 
1 
1 
3 
5 
6 





60 


60 


59 


5fl 


6 


6.0 


60 


5 9 


5. 


7 


7 





7 





6 


9 


6. 


8 


8 





8 





7 


9 


7. 


9 


9 


1 


9 





8 


9 


8- 


10 


10 


1 


10 





9 


9 


9- 


20 


20 


1 


20 





19 


8 


19. 


30 


30 


2 


30 





29 


7 


29. 


40 


40 


3 


40 





39 


6 


39. 


50 


50 


4 


50 





49 


6 


49. 





58 


58 


57 


57 


6 


5.8 


5.8 


5.7 


5.7 


7 


6 


8 


6 


7 


6 


7 


6 


6 


8 


7 


8 


7 


7 


7 


6 


7 


6 


9 


8 


8 


8 


7 


8 


6 


8 


5 


10 


9 


7 


9 


g 


9 


6 


9 


5 


20 


19 


5 


19 


3 


19 


1 


19 





30 


29 


2 


29 





28 


7 


28 


5 


40 


39 





38 


6 


38 


3 


38 





50 


48 


7 


48 


3 


47 


9 


47 


5 



10 
20 
30 
40 
50 



56 


56 


55 


55 


5.6 


5.6 


5-5 


5-5 


6 


6 


6 


5 


6 


5 


6 


4 


7 


5 


7 


4 


7 


4 


7 


3 


8 


5 


8 


4 


8 


3 


8 


2 


9 


4 


9 


3 


9 


2 


9 


1 


18 


8 


18 


6 


18 


5 


18 


3 


28 


2 


28 





27 


7 


27 


5 


37 


6 


37 


3 


37 





36 


5 


47 


1 


46 


6 


46 


2 


45 


8 





54 


3 


2 


6 


5.4 


0.3 


0. 


7 


6 


3 





3 


0. 


8 


7 


2 





4 


0. 


9 


8 


2 





4 


0. 


10 


9 


1 





5 





20 


18 


1 


1 





0. 


30 


27 


2 


1 


5 


1. 


40 


36 


3 


2 





1. 


50 


45 


4 


2 


5 


2. 



P. p. 



617 



77° 



13° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



166 



' Log. Sin, d. Log. Tan. c.d, Log. Cot. Log. Cos. 



O 

1 
2 
3 

5 
6 
7 
8 

_9^ 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 



9-35 209 
9.35 263 
9.35 318 
9.35 372 
9-35 427 



9.35 481 
9.35 536 
9.35 590 
9.35 644 
9. 35 698 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29 



30 

31 
32 
33 
34 
35 
36 
37 
38 
39 



40 

41 
42 
43 
44 
45 
46 
47 
48 
49^ 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59, 
60 



9.35 752 
9.35 806 
9.35 860 
9. 35 914 
9. 35 968 



9. 36 021 
9.36 075 
9-36 128 
9.36 182 
9.36 235 



9.36 289 
9.36 342 
9-36 395 
9.36 448 
9.36 501 



9.36 554 
9-36 607 
9.36 660 
9.36 713 
9-36 766 



9.36 818 
9.36 871 
9.36 923 
9. 36 976 
9-37 028 



9.37 081 
9.37 133 
9. 37 185 
37 237 
9.37289 



9-37 341 
9-37 393 
9-37 445 
9-37 497 
9-37 548 



9-37 600 
9-37 652 
9-37 703 
9-37 755 
9-37 806 



9-37 857 
9-37 909 
9-37 960 
9.38 011 
9.38 062 



9.38 113 
9.38 164 

38 215 
9.38 266 

38 317 



9-38 367 



Log. Cos, 



54 

54 

54 

54 

54 

54 

54 

54 

54 

54 

54 

54 

53 

54 

53 

53 

53 

53 

53 

53 

53 

53 

53 

53 

53 

53 

53 

52 

53 

52 

52 

52 

52 

52 

52 

52 

52 

52 

52 

52 

52 

51 

52 

51 

52 
51 
5l 
51 
5l 
51 
5l 
51 
5l 
51 
51 
51 
50 
51 
51 
50 



9-36 336 
9 36 394 
9. 36 451 
9.36 509 
9-36 566 



9-36 623 
936 681 
9-36 738 
9-36 795 
9-36 852 



9-36 909 
9.36 966 
9-37 023 
9-37 080 
9-37 136 



9-37 193 
9-37 250 
9-37 306 
9-37 363 
37 419 



-37 756 

-37 812 

9-37 868 

9. 37 924 

9.37 979 



9.37 475 
9. 37 532 
9-37 588 
9-37 644 
9-37 700 



9-38 035 
938 091 
9-38 146 
9-38 202 
9-38 257 



9-38 313 
938 368 
9.38 423 
9. 38 478 
9.38 533 



38 589 

38 644 

9.38 698 

9.38 753 

9. 38 808 



9-38 863 
9-38 918 

9. 38 972 

9.39 027 
9-39 081 



9-39 136 
9-39 190 
9-39 244 
9-39 299 
9-39 353 



9-39 407 
9-39 461 
9-39 515 
9-39 569 
9-39 623 



939 677 
Log. Cot. 



57 
57 
57 
57 
57 
57 
57 
57 
57 

57 
57 
56 
57 
56 

57 
56 
56 
56 
56 
56 
56 
56 
56 
56 
56 
55 
56 
56 
55 
56 
55 
55 
55 
55 
55 
55 
55 
55 
55 

55 
55 
54 
55 
55 
54 
55 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
54 
53 



0-63 663 
63 606 
0-63 548 
0-63 491 
0-63 433 



0-63 376 
0-63 319 
0-63 262 
0-63 204 
0-63147 



063 090 
0-63 033 
0-62 977 
0-62 920 
0-62 863 



0-62 806 
0-62 750 
0-62 693 
0-62 637 
0.62 580 



9. 98 872 
98 869 
9-98 866 
9.98 863 
9- 98 860 



9.98 858 
98 855 
9-98 852 
9-98 849 
9-98 846 



9- 98 843 

98 840 

98 837 

9.98 834 

9.98 831 



62 524 
62 468 
62 412 
62 356 
62 299 



0.62 243 
0-62 188 
0-62 132 
0-62 076 
0-62 020 



0-61 964 
0-61 909 
0-61 853 
0-61 798 
0.61 742 



0.61 687 
0.61 632 
0.61 576 
0.61 521 
0-61 466 



0.61 411 
0-61 356 
0-61 301 
0-61 246 
061 191 



98 828 
9-98 825 
9-98 822 
9-98 819 
998 816 



. 98 813 

998 810 

9-98 807 

98 804 

98 801 



9-98 798 
98 795 
9-98 792 
9. 98 789 
9.98 786 



9. 98 783 
9.98 780 
9.98 777 
9- 98 774 
9.98 771 



9. 98 768 
9. 98 765 
9.98 762 
9-98 759 
9-98 755 



0.61 137 
0-61 082 
0-61 027 
0-60 973 
0-60 918 



0-60 864 
0-60 809 
0-60 755 
0-60 701 
0-60 647 



0-60 592 
0-60 538 
0-60 484 
0-60 430 
060 376 



98 752 
9- 98 749 
9-98 746 
9-98 743 
9-98 740 



9- 98 737 
9-98 734 
9-98 731 
9.98 728 
9.98 725 



9-98 721 
9-98 718 
9. 98 715 
9-98 712 
9-98 709 



P. P. 



0-60 323 



Log. Tan. 



9. 98 706 
9- 98 703 
9-98 700 
9-98 696 
9- 98 693 



9-98 690 
Log. Sin. 





57 


57 


56 


6 


5-7 


5-7 


5-6 


7 


6 


7 


6 


6 


6 


6 


8 


7 


6 


7 


6 


7 


5 


9 


8 


6 


8 


5 


8 


5 


10 


9 


6 


9 


5 


9 


4 


20 


19 


1 


19 





18 




30 


28 




28 


5 


28 


2 


40 


38 


3 


38 





37 


6 


50 


47 


9 


47 


5 


47 


l-l 



56 

5-6 

6-5 

7-3 

8-4 

9-3 

18-6 

28-0 

37.3 

46.6 





55 


55 


54 


54 


6 


5-5 


5.5 


5.4 


5-4 


7 


6-5 


6 


4 


6.3 


6 


3 


8 


7-4 


7 


3 


7-2 


7 


2 


9 


8 3 


8 


2 


82 


8 


1 


10 


9-2 


9 


1 


9.1 


9 





20 


18-5 


18 


3 


18.1 


18 





30 


27-7 


27 


5 


27.2 


27 





40 


37-0 


36 


g 


36-3 


36 





50 


46.2 


45 


8 


45-4 


45 








53 


53 


53 


52 


6 


5-3 


5-3 


5-2 


5-2 


7 


6 


2 


6 


2 


6 


1 


6 


6 


8 


7 


1 


7 





7 





6 


9 


9 


8 





7 


9 


7 


9 


7 




10 


8 


9 


8 


g 


8 


7 


8 


f) 


20 


17 


8 


17 


6 


17 


5 


17 


^ 


30 


26 


7 


26 


5 


26 


2 


26 





40 


35 


6 


35 


3 


35 





34 


^ 


50 


44 


6 


44 


1 


43 


7 


43 


3 





^^. 


51 


50 


6 


5-1 


5-1 


5-6 


7 


6 





5 


9 


5 


9 


8 


6 


8 


6 


8 


6 


7 


9 


7 


7 


7 


6 


7 


6 


10 


8 


6 


8 


5 


8 


4 


20 


17 


1 


17 





16 


8 


30 


25 


7 


25 


5 


25 


2 


40 


34 


3 


34 





33 


Q 


50 


42 


9 


42 


5 


42 


1 





3 


3 


6 


0-3 


0.3 


7 


04 


0-3 


8 


0-4 


0-4 


9 


0-5 


0-4 


10 


0-6 


0-5 


20 


1.1 


1-0 


30 


1.7 


1-5 


40 


2-3 


2-0 


50 


2.9I 


2.5 



0.2 
0-3 
0-3 
0.4 
0-4 
0-8 
1.2 
1.6 
2.1 



P.P. 



103° 



618 



76'' 



TABLE VII. 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



165° 



Log. Sin. 



38 367 
38 418 
38 468 
38 519 
38 569 



38 620 
38 670 
38 720 
38 771 
38 821 



38 871 
38 921 

38 971 

39 021 
39 071 
39 120 
39 170 
39 220 
39 269 
39 319 



39 368 
39 418 
39 467 
39 516 
39 566 



39 615 
39 664 
39 713 
39 762 
39 811 



39 860 
39 909 

39 957 

40 006 
40 055 



40 103 
40 152 
40 200 
40 249 
40 297 



40 345 
40 394 
40 442 
40 490 
40 538 



40 586 
40 634 
40 682 
40 730 
40 777 



40 825 
40 873 
40 920 

40 968 

41 015 



41 063 
41 110 
41 158 
41 205 
41252 



41 2991 



Log. Tan. c. d. Log. Cot. Log. Cos 



39 677 
39 731 
39 784 
39 838 
39 892 



39 945 

39 999 

40 052 
40 106 
40 159 



40 212 
40 265 
40 318 
40 372 
40 425 



40 478 
40 531 
40 583 
40 636 
40 689 



40 742 
40 794 
40 847 
40 899 
40 952 



41 004 
41 057 
41 109 
41 16l 
41 213 



41 266 
41 318 
41 370 
41 422 
41 474 



41 525 
41 577 
41 629 
41 681 
41 732 



41 784 
41 836 
41 887 
41 938 
41 990 



42 041 
42 092 
42 144 
42 195 
42 246 



42 297 
42 348 
42 399 
42 450 
42 501 



42 552 
42 602 
42 653 
42 704 
42 754 



42 805 



Log. Cos.j d. jLog. Cot. c. d 



0.60 323 
0-60 269 
0-60 215 
0.60 16l 
0.60 108 



0.60 054 
0.60 001 
0-59 947 
0.59 894 
0-59 841 



0.59 
059 
0.59 
0-59 
0-59 



787 
734 
681 
628 
575 



0.59 
0.59 
0.59 
0.59 
0.59 



522 
469 
416 
363 
311 



0.59 258 
0.59 20 
0.59 153 
0.59 100 
0.59 048 



0-58 995 
0-58 943 
0-58 891 
0-58 838 
0.58 786 



0.58 734 
0.58 682 
0-58 630 
0.58 578 
0-58 52^ 



0.58 474 
0-58 422 
0.58 370 
0.58 319 
0.58 267 



0.58 216 
0.58 164 
0.58 112 
0.58 061 
0-58 010 



0.57 958 
0-57 907 
0.57 856 
0-57 805 
0-57 753 



0-57 702 
0-57 651 
0-57 600 
0-57 549 
0-57 499 



0-57 448 
0-57 397 
0.57 346 
0.57 296 
0.57 245 



0.57 195 



Log, Tan. 



98 690 
98 687 
98 684 
98 681 
98 678 



98 674 
98 671 
98 668 
98 665 
98 662 



98 658 
98 655 
98 652 
98 649 
98 646 



98 642 
98 639 
98 636 
98 633 
98 630 



98 626 
98 623 
98 620 
98 617 
98 613 



98 610 
98 607 
98 604 
98 600 
98 597 



98 594 
98 591 
98 587 
98 584 
98 581 



98 578 
98 574 
98 571 
98 568 
98 564 



98 561 
98 558 
98 554 
98 551 
98 548 



98 544 
98 54l 
98 538 
98 534 
98 53l 



98 528 
98 524 
98 521 
98 518 
98 514 



98 511 
98 508 
98 504 
98 501 
98 498 



98 494 



Log. Sin, 



P. P. 



54 

5.4 

6.3 

7.2 

8.1 

9.0 

18.0 

27.0 

36.0 

45.0 



53_ 

5.3 
2 
1 

9 



53 

5-3 
2 




53_ 

5.2 



52 

5.2 



51 

5.1 

6.0 

6 

7 

8 
17 
25 
34 
42 



51 

5.1 





50 


50 


49 


49 


6 


5.0 


5.0 


4.9 


4. 


7 


5 


9 


5 


8 


5 


8 


5. 


8 


6 


7 


6 


6 


6 


6 


6. 


9 


7 


6 


7 


5 


7 




7. 


10 


8 


4 


8 


3 


8 


2 


8. 


20 


16 


8 


16 


6 


16 


5 


16. 


30 


25 


2 


25 





24 


7 


24. 


40 


33 


6 


33 


3 


33 





32. 


50 


42 


1 


41 


6 


41 


2 


40. 





48 


48 


47 


47 


6 


4.8 


4.8 


4.7 


4 


7 


5 


6 


5 


6 


5 


5 


5 


8 


6 


4 


6 


4 


6 


3 


6. 


9 


7 


3 


7 


2 


7 


] 


7. 


10 


8 


1 


8 





7 




7. 


20 


16 


1 


16 





15 


3 


15. 


30 


24 


2 


24 





23 


7 


23. 


40 


32 


3 


32 





31 


(^ 


31. 


50 


40 


4 


40 





39 


6 


89. 



a 




a 


0.3 








4 








4 


0. 





5 


0. 





6 


0. 


1 


1 


1 


1 


7 


1. 


2 


3 


2. 


2 


9 


2. 



P. p. 



619 



75° 



TABLE VII.- 



15^ 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



164* 



7 

8 

J_ 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 



Log. Sin, 



9.41 299 
9.41346 
9.41 394 
41441 
9. 41 488 



9-41 534 
9.41 581 
9-41 628 
9.41 675 
9.41 721 



41 768 

41 815 

9.41861 

9.41 908 

9.41 954 



20 

21 
22 
23 
24 
25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



50 d 



51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



9.42 000 
9.42 047 
9-42 093 
9.42 139 
9-42 185 



9.42 232 
9.42 278 
9.42 324 
9.42 369 
9.42 415 



9.42 461 
9.42 507 
9.42 553 
9.42 598 
9.42 644 



42 690 
42 735 
42 781 
42 826 
42 87l 



42 917 

42 962 

43 007 
43 052 
43 098 



43 143 
43 188 
43 233 
43 278 
43 322 



9.43 367 
9.43 412 
9.43 457 
9.43 50l 
9-43 546 



.43 591 
9.43 635 
9.43 680 
9.43 724 
9-43 768 



9.43 813 
9.43 857 
9.43 901 
9.43 945 
9-43 989 



9-44 034 



Log. Cos. 



Log. Tan, c. d. Log. Cot. Log. Cos 



42 805 
42 856 
42 906 

42 956 

43 007 



43 057 
43 107 
43 157 
43 208 
43 258 



43 308 
43 358 
43 408 
43 458 
43 508 



43 557 
43 607 
43 657 
43 706 
43 756 



43 806 
43 855 
43 905 

43 954 

44 003 



44 053 
44 102 
44 151 
44 200 
44 249 



44 299 
44 348 
44 397 
44 446 
44 494 



44 543 
44 592 
44 641 
44 690 
44 738 



44 787 
44 835 
44 884 
44 932 
44 981 



45 029 
45 077 
45 126 
45 174 
45222 




9-45 749 
,og. Cot. 



51 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 
49 
50 
49 
49 
50 
49 
49 
49 
49 
49 

49 
49 
49 
49 
49 

49 
49 
49 
49 
48 
49 
49 
48 
49 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
47 
48 
48 
47 
48 
47 



57 195 
57 144 
57 094 
57 043 
56 993 



56 942 
56 892 
56 842 
56 792 
56 742 



56 692 
56 642 
56 592 
56 542 
56 492 



56 442 
56 392 
56 343 
56 293 
56 243 



56 194 9 
56 144 
56 095 
56 045 
55 996 



55 947 
55 898 
55 848 
55 799 
55 750 



55 701 
55 652 
55 603 
55 554 
55 505 



55 456 
55 407 
55 359 
55 310 
55 261 



0.55 213 
0.55 164 
055 116 
0.55 067 
0.55 019 



0.54 970 
0.54 922 
0.54 874 
0.54 825 
0.54 777 



0.54 729 
0.54 681 
0.54 633 
0.54 585 
0.54 537 



0.54 489 
. 54 441 
0.54 393 
0.54 346 
054 298 



0. 54 250 



Log. Tan, 



98 494 
98 491 
98 487 
98 484 
98 481 

98 477 
98 474 
98 470 
98 467 
98 464 



98 460 
98 457 
98 453 
98 450 
98 446 



98 443 
98 439 
98 436 
98 433 
98 429 



98 426 
98 422 
98 419 
98 415 
98 412 



98 408 
98 405 
98 401 
98 398 
98 394 



98 391 
98 387 
98 384 
98 380 
98 377 



98 373 
98 370 
98 366 
98 363 
98 359 



98 356 
98 352 
98 348 
98 345 
98341 



98 338 
98 334 
98 331 
98 327 
98 324 




9-98 284 
Log. Sin. 



60 

59 
58 
57 
56. 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46_ 
45 
44 
43 
42 
41 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26. 
25 
24 
23 
22 
21. 
30 
19 
18 
17 
16. 
15 
14 
13 
12 
11 
10 

9 

8 

7 
_6^ 

5 

4 

3 

2 

O 



P.P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



50 



5.0 


5 


5.9 


5. 


6-7 


6 


7.6 


7. 


8.4 


8 


16.8 


16 


25.2 


25. 


33.6 


33. 


42.1 


41. 



50 


8 
6 
5 
3 
6 

3 
6 





49 


49 


48 


48 


6 


4.9 


4.9 


4-8 


4.8 


7 


5.8 


5.7 


5.6 


5 


6 


8 


6.6 


6.5 


6.4 


6 


4 


9 


7.4 


7.3 


7.3 


7 


2 


10 


8.2 


8.1 


8.1 


8 





20 


16.5 


16.3 


16.1 


16 





30 


24.7 


24.5 


24.2 


24 





40 


33.0 


32.6 


32.3 


32 





50 


41.2 


40.8 


40.4 


40 








47 


47 


46 


4( 


6 


4.7 


4.7 


4.6 


4. 


7 


5 


5 


5.5 


5.4 


5. 


8 


6 


3 


6.2 


6.2 


6 


9 


7 


1 


7.0 


7.0 


6 


10 


7 


9 


7.8 


7.7 


7 


20 


15 


8 


15.6 


15.5 


15. 


30 


23 


7 


23.5 


23.2 


23. 


40 


31 


G 


31.3 


31.0 


30. 


50 


39 


6 


39.1 


38-7 


38. 





45 


45 


41 


6 


4.5 


4-5 


4.4 


7 


5 


3 


5-2 


5.2 


8 


6 





6.0 


5.9 


9 


6 


8 


6.7 


6.7 


10 


7 


6 


7.5 


7.4 


20 


15 


1 


15.0 


14.8 


30 


22 


7 


22.5 


22.2 


40 


30 


3 


30.0 


29.6 


50 


37 


9 


37.5 


37.1 



44 

4.4 

5.1 

5.8 

6.6 

7.3 

14.6 

22. 

29.3 

36.6 





4 


3 


3 


6 


0.4 


0.3 


0.3 J 


7 


0-4 


0.4 





3 i 


8 


0.5 


0.4 







9 


0.6 


0.5 





4 


10 


0.6 


0.6 





5 


20 


1.3 


1.1 


1 





30 


2.0 


1.7 


1 


5 


40 


2.6 


2.3 


2 





50 


3.3 


2.9 


2 


5 . 

1 



P.P. 



105** 



620 



74* 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGENTS, 
16° AND COTANGENTS. 



163" 



Log. Sin. d. Log. Tan. c. d. Log. Cot. Log. Cos. d 



44 034 
44 078 
44 122 
44 166 
44 209 



44 253 
44 297 
44 341 
44 384 
44 428 



44 472 
44 515 
44 559 
44 602 
44 646 



44 689 
44 732 
44 776 
44 819 
44 862 



44 905 
44 948 

44 991 

45 034 
45 077 



45 120 
45 163 
45 206 
45 249 
45 291 



45 334 
45 377 
45 415 
45 462 
45504 



45 547 
45 589 
45 631 
45 674 
45 716 



45 758 
45 800 
45 842 
45 885 
45 927 



45 969 

46 011 
46 052 
46 094 
46 136 



46 178 
46 220 
46 261 
46 303 
46 345 



46 386 
46 428 
46 469 
46 511 
46 552 



9-46 593 



Log. Cos. 



46 224 
46 271 
46 318 
46 366 
46 413 



45 749 
45 797 
45 845 
45 892 
45 940 



45 987 

46 035 
46 082 
46 129 
46 177 



46 460 
46 507 
46 554 
46 601 
46 647 



46 694 
46 741 
46 788 
46 834 
46 881 
46 928 

46 974 

47 021 
47 067 
47 114 



47 160 
47 207 
47 253 
47 299 
47 345 



47 392 
47 438 
47 484 
47 530 
47 576 



47 622 
47 668 
47 714 
47 760 
47 806 



47 851 
47 897 
47 943 

47 989 

48 034 



48 080 
48 125 
48 171 
48 216 
48 262 



48 307 
48 353 
48 398 
48 443 
48 488 



9-48 534 1 
Log. Cot. 



0.54012 
0.53 965 
0.53 917 
0.53 870 
0-53 823 



0.53 776 
0.53 728 
0.53 681 
0.53 634 
0.53 587 



0.53 540 
0.53 493 
0.53 446 
53 399 
0-53 352 



54 250 
54 202 
54 155 
54 107 
54 060 



0.53 305 
0.53 258 
0.53 212 
0.53 165 
0.53 118 



53 072 
53 025 
52 979 
52 932 
52 886 



52 839 
52 793 
52 747 
52 700 
52 654 



52 608 

52 562 

0-52 516 

0-52 469 

0-52 428 



0.52 377 
0-52 331 
0.52 286 
0.52 240 
0.52194 



0.52 148 
0.52 102 
0.52 057 
0.52 Oil 
0.51 965 



0.5] 920 
0-51 874 
0.51 829 
0.51 783 
0.51 738 



C.d. 



0.51 692 
0.51 647 
0.51 602 
0.51 556 
0.51 511 



0.51 466 



Log. Tan 



98 284 
98 280 
98 277 
98 273 
98 269 



98 266 
98 262 
98 258 
98 255 
98 251 



98 247 
98 244 
98 240 
98 236 
98 233 



98 229 
98 225 
98 222 
98 218 
98 214 



98 211 
98 207 
98 203 
98 200 
98 196 




98 155 
98 151 
98 147 
98 143 
98 140 



98 136 
98 132 
98 128 
98 124 
98 121 



98 117 
98 113 
98 109 
98 105 
98 102 



98 098 
98 094 
98 090 
98 086 
98 082 



98 079 
98 075 
98 071 
98 067 
98 063 



98 059 



Log. Sin. 



60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
'14 
13 
12 
11 
10 
6 
8 
7 
6 
5 
4 
3 
2 
1 





P. P. 





48 


47 


41 


6 


4.8 


4.7 


4. 


7 


5 


6 


5 


5 


5. 


8 


6 


4 


6 


3 


6. 


9 


7 


2 


7 


1 


7. 


10 


8 





7 


9 


7- 


20 


16 





15 


8 


15. 


30 


24 





23 


7 


23. 


40 


32 





31 


6 


31. 


50 


40 





39 


6 


39. 





46 


46 


45 


45 


6 


4.6 


4.6 


4.5 


4. 


7 


5=4 


5.3 


5 


3 


5. 


8 


6-2 


6.1 


6 





6. 


9 


7.0 


6.9 


6 


8 


6. 


10 


7.7 


7.6 


7 


6 


7. 


20 


15.5 


15.3 


15 


1 


15. 


30 


23.2 


23.0 


22 


7 


22. 


40 


31-0 


30.6 


30 


3 


30. 


50 


38.7 


38.3 


37 


9 


37. 





44 


43 


4. 


6 


4.4 


4.3 


4 


7 


5 


1 


5.1 


5 


8 


5 


8 


5.8 


5 


9 


6 


6 


6.5 


6 


10 


7 


3 


7.2 


7 


20 


14 


6 


14.5 


14 


30 


22 





21.7 


21 


40 


29 


3 


29.0 


28 


50 


36 


6 


36.2 


35 



43 

4.2 

4 

5 

6 

7 
14 
21 
28 
35 



43 

4.2 

4.9 

5.6 

6.3 

7.0 

14.0 

21.0 

28.0 

35.0 



41 

4.1 



41 

4 

4 

5 

6 

6 
13 
20 
27 
34 






4 





4 





5 





6 





6 


1 


3 


2 





2 


6 


3 


3 



3 

0.3 
0.4 
0.4 
0.5 
0.6 

11 
1-7 
2.3 
2.9 



P.P. 



621 



73'' 



17° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



163** 



' Log> Sin. d. Log. Tan. c. d. Log. Cot. Log. Cos. d. 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 
20 
21 
22 
23 
2i 
25 
26 
27 
28 
29. 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 



60 

51 
52 
53 
54 
55 
56 
57 
58 
59_ 

fiO 



9.46 593 
9.46 635 
9.46 676 
46 717 
9.46 758 



9.46 799 
9.46 840 
9.46 881 
9.46 922 
9.46 963 



47 004 
9.47 045 
9-47 086 

47 127 
9.47 168 



9.47 208 
9.47 249 
9.47 290 
9.47 330 
9-47 371 



47 411 
9-47 452 
9.47 492 
9.47 532 
9.47 573 



9-47 613 
9.47 653 
9.47 694 
9.47 734 
9.47 774 



9.47 814 
9.47 854 
9.47 894 
9-47 934 
9.47 974 



9.48 014 
9.48 054 
9.48 093 
9.48 133 
9.48 173 



9.48 213 
9.48 252 
9.48 292 
9.48 331 
9. 48 371 



48 410 

48 450 

9.48 489 

9.48 529 

9. 48 568 



9.48 607 
9.48 646 
9.48 686 
9.48 725 
48 764 



9.48 803 
48 842 
9.48 881 
9.48 920 
9.48 959 



9-48 998 



48 534 
48 579 
48 624 
48 669 
48 714 



48 759 
48 804 
48 849 
48 894 
48 939 



48 984 

49 028 
49 073 
49 118 
49 162 



49 207 
49 252 
49 296 
49 341 
49 385 



49 430 
49 474 
49 518 
49 563 
49 607 



49 651 
49 695 
49 740 
49 784 
49 828 



49 872 
49 916 

49 960 

50 004 
50 048 



50 092 
50 136 
50 179 
50 223 
50 267 



50 311 
50 354 
50 398 
50 442 
50 485 



50 529 
50 572 
50 616 
50 659 
50 702 



50 746 
50 789 
50 832 
50 876 
50 919 



50 962 

51 005 
51 048 
51 091 
51 134 



51 177 



0.51 486 
0.51421 
0.51 376 
0.51 330 
0-51 285 



0-51 240 
0.51 195 
0-51 151 
0-51 106 
051 061 



0-51 016 
0.50 971 
0.50 926 
0.50 882 
0-50 837 



0-50 
0-50 
0.50 
0.50 
0.50 



792 
748 
703 
659 
614 



0.50 
0.50 
0.50 
0.50 
0.50 



570 
525 
481 

437 
392 



0-50 348 
0.50 304 
0.50 260 
0.50 216 
0.50 172 



0-50 128 
0-50 083 
0-50 039 
0-49 996 
0-49 952 



0-49 
0.49 
0.49 
0.49 
0.49 



908 
864 
820 
776 
733 



0.49 
0-49 
0-49 
0.49 
0.49 



689 
645 
602 
558 
514 



0.49 471 
0.49 427 
0-49 384 
0-49 340 
0-49 297 



0-49 
0-49 
0-49 
0-49 
0-49 



0.49 
0.48 
0.48 
0-48 
0-48 
0.48 



254 
210 
167 
124 
081 
038 
994 
951 
908 
865 
822 




98 059 
98 056 
98 052 
98 048 
98 044 



98 040 
98 036 
98 032 
98 028 
98 024 



98 001 
97 997 
97 993 
97 989 
97 985 



97 981 
97 977 
97 973 
97 969 
97 966 



97 962 
97 958 
97 954 
97 950 
97 946 



97 942 
97 938 
97 934 
97 930 
97 926 



97 922 
97 918 
97 914 
97 910 
97 906 



97 902 
97 898 
97 894 
97 890 
97 886 



97 881 
97 877 
97 873 
97 869 
97 865 



97 861 
97 857 
97 853 
97 849 
97 845 



97 841 
97 837 
97 833 
97 829 
97 824 



97 820 



P.P. 





45 


45 


44 


44 


6 


4-5 


4-5 


4-4 


4.4 


7 


5 


3 


5 


2 


5-2 


5 


1 


8 


6 





6 





5.9 


5 


8 


9 


6 


8 


6 


7 


6-7 


6 


g 


10 


7 


6 


7 


5 


7-4 


7 


3 


20 


15 


1 


15 





14-8 


14 


Q 


30 


22 


7 


22 


5 


22-2 


22 


Q 


40 


30 


3 


30 





29-6 


29 


3 


50 


37 


9 


37 


5 


37.1 


36 


6 



43 43 


6 


4.3 


4.3 


7 


5 


1 


5 





8 


5 


8 


5 


7 


9 


6 


5 


6 


4 


10 


7 


2 


7 


1 


20 


14 


5 


14 


3 


30 


21 


7 


21 


5 


40 


29 





28 


5 


50 


36 


2 


35 


^ 





41 


41 


40 


40 


6 


4-1 


4-1 


4-0 


4-0 


7 


4 


8 


4 


8 


4 


7 


46 


8 


5 


5 


5 


4 


5 


4 


5-3 


9 


6 


2 


6 


1 


6 


1 


6-a 


10 


6 


9 


6 


8 


6 


7 


6-6 


20 13 


8 


13 


6 


13 


5 


13-3 


3020 


7 


20 


5' 


20 


2 


200 


40i27 


6 


27 


3 


27 





26 6 


50 


34 


6 


34 


1 


33 


7 33 3 

















39 39 38 


6 


3-9 


3-9 


38 


7 


4 


6 


4 


5 


4 


5 


8 


5 


2 


5 


2 


5 


1 


9 


5 


9 


5 


3 


5 


8 


10 


6 


6 


6 


5 


6 


4 


20 


13 


1 

J- 


13 





12 


8 


30 


19 


7 


19 


5 


19 


2 


40 


26 


3 


26 





25 


6 


50 


32 


9 


32 


5 


32 


1 



10 
20 
30 
40 
50 



4 3 

0.40-3 
40. 

5 0- 

6 0- 
6 0- 
3 1- 
1. 
6 2- 
3 2- 



Loc:. Cos, 



d. Log. Cot. 



c.d. 



Log. Tan.lLog. Sin. d 



P.P. 



107** 



622 



73* 



TABLE VII.- 



18° 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



161° 



' Log. Sin. d. Log. Tan. c, d. Log. Cot. Log. Cos. d 



48 998 

49 037 
49 076 
49 114 
49 153 



49 192 
49 231 
49 269 
49 308 
49 346 



49 385 
49 423 
49 462 
49 500 
49 539 



49 577 
49 615 
49 653 
49 692 
49 730 



49 768 
49 806 
49 844 
49 882 
49 920 



958 
996 
034 
072 
110 
147 
185 
223 
260 
298 
336 
373 
411 
448 
486 
523 
561 
598 
635 
672 
710 
747 
784 
821 
858 
895 
932 



006 
043 
080 
117 
154 
190 
227 
9-51 264 
Cos 



Log. 



177 
220 
263 
306 
349 



435 
477 
520 
563 
605 
648 
691 
733 
776 
818 
861 
903 
946 



9. 53 



Log. 



030 
073 
115 
157 
199 
241 
284 
326 
368 
410 
452 
494 
536 
578 
619 
661 
703 
745 
787 
828 
870 
912 
953 
995 
036 
078 
119 
161 
202 
244 
285 
326 
368 
409 
450 
491 
533 
574 
615 
656 
697 
Cot. 



0.48 822 
0.48 779 
0.48 736 
0.48 693 
0.48 650 



0-48 608 
0.48 565 
0.48 522 
0.48 479 
0.48 437 



0.48 394 
0.48 35l 
0.48 309 
0.48 266 
0.48 224 



0.48 181 
0.48 139 
0.48 096 
0.48 054 
0-48 012 



0.47 
0.47 
0.47 
0.47 
0.47 



0.47 
0.47 
0.47 
0.47 
0.47 



969 
927 
885 
842 
800 
758 
716 
674 
632 
590 



0.47 548 
0.47 506 
0.47 464 
0.47 422 
0.47 3Z0 



0.47 338 
0.47 296 
0.47 255 
0.47 213 
0-47 17l 



0.47 130 
0-47 088 
0.47 046 
0.47 005 
0-46 963 



0.46 922 
0.46 880 
0.46 839 
0.46 797 
0.46 756 



0.46 714 
0.46 673 
0.46 632 
0.46 591 
0-46 549 



0.46 508 
0.46 467 
0.46 426 
0.46 385 
0.46 344 



0.46 303 



Log, Tan. 



97 820 
97 816 
97 812 
97 808 
97 804 



97 800 
97 796 
97 792 
97 787 
97 783 



97 779 
97 775 
97 771 
97 767 
97 763 



97 758 
97 754 
97 750 
97 746 
97 742 



97 737 
97 733 
97 729 
97 725 
97 721 



97 716 
97 712 
97 708 
97 704 
97 700 



97 695 
97 691 
97 687 
97 683 
97 678 



97 674 
97 670 
97 666 
97 661 
97 657 



97 653 
97 649 
97 644 
97 640 
97 636 



97 632 
97 62 
97 623 
97 619 
97 614 



97 610 
97 606 
97 601 
97 597 
97 593 



97 588 
97 584 
97 580 
97 575 
97 571 



9-97 567 



108** 



Log, Sin, 
623 



P. P. 





43 


42 


6 


4.3 


4.2 


7 


5 





4 


9 


8 


5 


7 


5 


6 


9 


6 


4 


6 


4 


10 


7 


1 


7 


1 


20 


14 


3 


14 


1 


30 


21 


5 


21 


2 


40 


28 


6 


28 


3 


50 


35 


8 


35 


4 



41 

4.1 



43 

4.2 

4.9 

5.6 

6.3 

7.0 

14.0 

21.0 

28.0 

35. 



41 

4.1 





39 


38 


6 


3.9 


3.8! 


7 


4 


5 


4 


5 


8 


5 


2 


5 


I 


9 


5 


8 


5 


8 


10 


6 


5 


6 


4 


20 


13 





12 


g 


30 


19 


5 


19 


2 


40 


26 





25 


6 


50 


32 


5 


32 


1 



38 

38 

4.4 

5.0 

5.7 

6.3 

i 12.6 

i 19-0 

125.3 

.31.6 



37_ 

3.7 

4.4 

5.0 

5.6 

6.2 

12.5 

18.7 

25.0 

31.2 



37 

3.7 

4 

4 

5 

6 
12 
18 
24 .. 
30. § 





^ 


6 


0.4 


7 


0.5 


8 


0.6 


9 


0.7 


10 


0.7 


20 


1.5 


30 


2.2 


40 


3.0 


50 


3.7 



36 

3.6 

4.2 

48 

5.5 

6.1 

12.1 

18. 2 

24.3 

30.4 



P. P. 



71** 



lO'' 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



160° 



' Log. Sin, d, Log. Tan. c.d. Log. Cot. Log. Cos. d 



P. P. 




51 629 
51 665 
51 702 
51 738 
51 774 



51 810 
51 847 
51 883 
51 919 
51 955 



51 991 

52 027 
52 063 
52 099 
52 135 



52 170 
52 206 
52 242 
52 278 
52 314 



52 349 
52 385 
52 421 
52 456 
52 492 



52 527 
52 563 
52 598 
52 634 
52 669 



52 704 
52 740 
52 775 
52 810 
52 848 



52 881 
52 916 
52 951 

52 986 

53 021 



53 056 
53 091 
53 126 
53 161 
53 196 



53 231 
53 266 
53 301 
53 335 
53 370 



9-53 405 



53 697 
9.53 738 
9.53 779 
9.53 820 

53 861 



9.53 902 
9.53 943 

9.53 983 

9.54 024 
54 065 



9.54 106 
9-54 147 
9.54 187 
9.54 228 
9.54 269 



9.54 309 
9.54 350 
9.54 390 
9.54 431 
9.54 471 



9.54 512 
9-54 552 
9.54 593 
9-54 633 
9.54 673 



9.54 714 
9-54 754 
9-54 794 
9-54 834 
9-54 874 



9-54915 
9-54 955 
9-54 995 
9-55 035 
9-55075 



9-55 115 
9-55 155 
9-55 195 
9-55 235 
9-55 275 



9.55 315 
9.55 355 
9-55 394 
55 434 
9-55474 



9-55 514 
9-55 553 
55 593 
9-55 633 
9.55 672 



9-55 712 
9-55 751 
55 791 
9-55 831 
9. 55 870 



9-55 909 
9. 55 949 
9. 55 988 
9. 56 028 
9. 56 067 



9-56 106 



Log. Cos, 



Log. Cot. 



39 
39 
39 
39 



46 303 
46 262 
46 221 
46 180 
46 139 



46 098 
46 057 
46 016 
45 975 
45 934 



45 894 
45 853 
45 812 
45 772 
.45 731 



.45 690 
.45 650 
45 609 
45 569 
45 528 



45 488 
45 447 
45 407 
45 367 
45 326 



45 286 
45 246 
45 205 
45 165 
45 125 



45 085 
45 045 
45 005 
44 965 
44 925 



44 884 
44 845 
44 805 
44 765 

44 725 



44 685 
44 645 
44 605 
44 565 
44 526 




0.44 090 
0.44 051 
0.44 011 
0.43 972 
0-43 932 



0.43 893 
Log. Tan. 



97 567 
97 562 
97 558 
97 554 
97 549 



97 545 
97 541 
97 536 
97 532 
97 527 



97 523 
97 519 
97 514 
97 510 
97 505 



97 501 
97 497 
97 492 
97 488 
97 483 



97 479 
97 475 
97 470 
97 466 
97 461 



97 457 
97 452 
97 448 
97 443 
97 439 



97 434 
97 430 
97 425 
97 421 
97 416 



97 412 
97 407 
97 403 
97 398 
97 394 



97 389 
97 385 
97 380 
97 376 
97 371 



97 367 
97 362 
97 358 
97 353 
97 349 



97 344 
97 340 
97 335 
97 330 
97 326 



97 321 
97 317 
97 312 
97 308 
97 303 



97 298 



_og. Sin. 



60 

59 
58 
57 
56 
55 
54 
53 
52 
_51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
j41 

40 

39 
38 
37 
_36 

35 
34 
33 
32 
11 
30 
29 
28 
27 
It 
25 
24 
23 
22 
21 

20 

19 
18 

17 
16. 
15 
14 
13 
12 

n 

10 

9 



6 

7 

8 

9 

10 

20 

30 

40 

50 



41 

4.1 

4.8 

5.4 

6.1 

6.8 

13.6 

20.5 

27.3 

34.1 



40 

4.0 

4.7 

5.4 

6.1 

6.7 

13.5 

20.2 

27.0 

33-7 



40 

4-0 





39 


6 


3.9 


7 


4.6 


8 


5.2 


9 


5-9 


10 


6-6 


20 


13-1 


30 


19-7 


40 


26-3 


50 


32.9 



39 

3-9 

4.5 

5-2 

5-8 

6-5 

13-0 

19-5 

26-0 

32.5 





37 


36 


6 


3-7 


3-6 


7 


4-3 


4.2 


8 


4-9 


4-8 


9 


5-5 


5-5 


10 


6-1 


6-1 


20 


12-3 


12-1 


30 


18-5 


18-2 


40 


24-6 


24-3 


50 


30-8 


30-4 





35 


35 


6 


3-5 


3-5 


7 


4 


1 


4-1 


8 


4 


7 


4-6 


9 


5 


3 


5-2 


10 


5 


9 


5-8 


20 


11 


8 


11-6 


30 


17 


7 


17-5 


40 


23 


6 


23-3 


50 


29 


6 


29.1 





S 




1 


6 


0-5 


0.4 


7 





6 





5 


8 





6 





6 


9 





7 





7 


10 





8 





7 


20 


1 


g 


1 


5 


30 


2 


5 


2 


2 


40 


3 


3 


3 





50 


4 


1 


3 


7 



36 

3.6 

4.2 

4.8 

5.4 

6.0 

12.0 

18.0 

24-0 

30-0 



34 

3-4 

4-0 

4-6 

5.2 

5.7 

11.5 

17-2 

23-0 

28.7 



4 

0-4 
0-4 
0.5 
0-6 
0-6 
1-3 
2.0 
2-6 
3-3 



P.P. 



109'' 



624 



70** 



20° 



TABLE VIl— 1.0GARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



159° 



Log. Sin, 



9 



53 405 
53 440 
53 474 
53 509 
53 544 



53 578 
53 613 
53 647 
53 682 
53 716 



53 750 
53 785 
53 819 
53 854 
53 888 



53 922 
53 956 

53 990 

54 025 
54 059 



54 093 
54 127 
54 161 
54 195 
54 229 



54 263 
54 297 
54 331 
54 365 
54 398 



54 432 
54 466 
54 500 
54 534 
54 567 



9-55 
Log. 



601 
634 
668 
702 
735 
769 
802 
836 
869 
902 
936 
969 
002 
036 
069 
102 
135 
168 
202 
235 
268 
301 
334 
367 
400 
433 
Cos 



Log. Tan, 



c.d, 



56 106 
56 146 
56 185 
56 224 
56 263 



56 303 
56 342 
56 381 
56 420 
56 459 



56 498 
56 537 
56 576 
56 615 
56 654 



56 693 
56 732 
56 771 
56 810 
56 848 



56 887 
56 926 

56 965 

57 003 
57 042 



57 081 
57 119 
57 158 
57 196 
57 235 



57 274 
57 312 
57 350 
57 389 
57 427 



57 466 
57 504 
57 542 
57 581 
57 619 



57 657 
57 696 
57 734 
57 772 
57 810 



57 848 
57 886 
57 925 

57 963 

58 001 




Log. Cot, 



38 



Log. Cot. Log. Cos, 



43 893 
43 854 
43 815 
43 775 
43 736 



43 697 
43 658 
43 619 
43 580 
43 540 



43 501 
43 462 
43 423 
43 384 
43346 



43 307 
43 268 
43 229 
43 190 
43 151 



43 112 
43 074 
43 035 
42 996 
42 958 



42 919 
42 880 
42 842 
42 803 
42 765 



42 726 
42 687 
42 649 
42 611 
42^72 



42 534 
42 495 
42 457 
42 419 
42 380 



42 342 
42 304 
42 266 
42 227 
42 189 



42 151 
42 113 
42 075 
42 037 
41 999 



41 961 
41 923 
41 885 
41 847 
41 809 



41 771 
41 733 
41 695 
41 658 
41 620 



0-41 582 
Log. Tan. 



97 298 
97 294 
97 289 
97 285 
97 280 



97 275 
97 271 
97 266 
97 261 
97 257 



97 252 
97 248 
97 243 
97 238 
97 234 



97 229 
97 224 
97 220 
97 215 
97 210 



97 206 
97 201 
97 196 
97 191 
97 187 



97 182 
97 177 
97 173 
97 168 
97 163 



97 159 
97 154 
97 149 
97 144 
97 140 



97 135 
97 130 
97 125 
97 121 
97 116 



97 111 
97 106 
97 102 
97 097 
97 092 



97 087 
97 082 
97 078 
97 073 
97 068 



97 063 
97 058 
97 054 
97 049 
97 044 



97 039 
97 034 
97 029 
97 025 
97 020 



9-97 015 
Log. Sin. 



60 

59 
58 
57 
56, 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 

25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
_6 
5 



P. P. 





39 


3f 


6 


3 9 


3. 


7 


4 


6 


4. 


8 


5 


2 


5. 


9 


5 


9 


5. 


10 


6 


6 


6. 


20 


13 


1 


13. 


30 


19 




19. 


40 


26 


3 


26. 


50 


32 


9 


32. 





38 


38 , 


6 


3.8 


3.8| 


7 


4 


5 


4 


4 


8 


5 


1 


5 





9 


5 


8 


5 


7 


10 


6 


4 


6 


3 


20 


12 


8 


12 


6 


30 


19 


2 


19 





40 


25 


6 


25 


3 


50 


32 


1 


31 


6 





35 


34 


34 


6 


3.5 


3.4 


3. 


7 


4.1 


4.0 


3. 


8 


4.6 


4.6 


4. 


9 


5.2 


5.2 


5. 


10 


5.8 


5.7 


5. 


20 


11.6 


11.5 


11. 


30 


17.5 


17.2 


17. 


40 


23.3 


23.0 


22. 


50 


29.1 


28.7 


28. 



37_ 

3.7 

4.4 

5.0 

5.6 

6.2 

12.5 

18.7 

25-0 

31.2 





33- 


6 


3.31 


7 


3 


9 


8 


4 


4 


9 


5 





10 


5 


6 


20 


11 


1 


30 


16 


7 


40 


22 


3 


50 


27 


9 



33 

3.3 

3.8 

4.4 

4.9 

5.5 

11.0 

16.5 

22.0 

27.5 



? 

0.4 
0-5 
0.6 
0.7 
0.7 
1.5 
2.2 
3.0 
37 



P. P. 



110° 



62a 



69° 



31*^ 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



158' 



55 433 
55 466 
55 498 
55 531 
55 564 



55 597 
55 630 
55 662 
55 695 
55 728 



55 760 
55 793 
55 826 
55 858 
55 891 



55 923 
55 956 

55 988 

56 020 
56 053 



56 085 
56 118 
56 150 
56 182 
56 214 




56 567 
56 599 
56 63l 
56 663 
56 695 



56 727 
56 758 
56 790 
56 822 
56 854 



56 885 
56 917 
56 949 

56 980 

57 012 



57 043 
57 075 
57 106 
57 138 
57 169 



57 201 
57 232 
57 263 
57 295 
57 326 



9-57 357 



Log. Cos. 



Log. Sin. d. iLog. Tan. c. d. Log. Cot. Log. Cos 



58 417 
58 455 
58 493 
58 531 
58 568 



58 606 
58 644 
58 68l 
58 719 
58 756 



58 794 
58 83l 
58 869 
58 906 
58 944 



58 981 

59 019 
59 056 
59 093 
59 131 



59 168 
59 205 
59 242 
59 280 
59 317 



59 354 
59 391 
59 428 
59 465 
59 502 



59 540 
59 577 
59 614 
59 651 
59 688 



59 724 
59 76l 
59 798 
59 835 
59 872 



59 909 
59 946 

59 982 

60 019 
60 056 



60 093 
60 129 
60 166 
60 203 
60 239 



60 276 
60 312 
60 349 
60 386 
60 422 



60 459 
60 495 
60 531 
60 568 
60 604 



60 641 



Log. Cot. c. d. 




41 582 
41 544 
41 507 
41 469 
41 431 



41 394 
41 356 
41 318 
41 281 
41 243 9 



206 
168 
131 
093 
056 
018 
981 
944 
906 
889 



40 460 
40 423 
40 386 
40 349 
40 312 



275 
238 
20l 
164 
128 



091 
054 
017 
980 
944 



39 907 
39 870 
39 833 
39 797 
39 760 



39 724 
39 687 
39 650 
39 614 
39 577 




Log, Tan. 



97 015 
97 010 
97 005 
97 000 
96 995 



96 991 
96 986 
96 981 
96 976 
96 971 



96 966 
96 961 
96 956 
96 952 
96 947 



96 942 
96 937 
96 932 
96 927 
98 922 



96 917 
96 912 
96 907 
96 902 
96 897 



96 892 
96 887 
96 882 
96 877 
96 873 



96 868 
96 863 
96 858 
96 853 
96 848 



96 843 
96 838 
96 833 
96 828 
96 823 



96 818 
96 813 
96 808 
96 802 
96 797 



96 792 
96 787 
96 782 
96 777 
96 772 



96 76Z 
96 762 
96 757 
96 752 
96 747 



96 742 
96 737 
96 732 
96 727 
96 72 1 
96 7161 



iir 



Log. Sin. d. 
626 



P. P. 





38 


37 


37 


6 


3.8 


3-7 


3.7 


7 


4 


4 


4.4 


4 


f^ 


8 


5 





5.0 


4 


9 


9 


5 


7 


5.6 


5 


5 


10 


6 


3 


6.^ 


6 


1 


20 


12 


6 


12.5 


12 


3 


30 


19 





18.7 


18 


5 


40 


25 


3 


25.0 


24 


5 


50 


31 


6 


31.2 


30 


8 





36 


36 


6 


3.6 


36 


7 


4 


2 


4 


2 


8 


4 


8 


4 


8 


9 


5 


5 


5 


4 


10 


6 


1 


6 





20 


12 


1 


12 





30 


18 


2 


18 





40 


24 


3 


24 





50 


30 


4 


30 


















33. 32 ae 


6 


3.3 


3.2 


3.2 


8 


3.8 


3 


8 


3 


7 


7 


4.4 


4 


3 


4 


2 


9 


4.9 


4 


9 


4 


8 


10 


5.5 


5 


4 


5 


3 


20 


11.0 


10 


8 


10 


Q 


30 


16.5 


16 


2 


16 





40 


22.0 


21 


6 


21 


3 


50 


27.5 


27.1 


26.6 




3 


T. 


3 


1 





6 

7 

8 

9 

10 

20 

30 

40 

50 



3 

4 

4 

5 
10 
15 

21.0 
26.2 





5 


5 


6 


0.5 


0.5 


7 





6 


0-6 


8 





7 


0.6 


9 





8 


0-7 


10 





9 


0.8 


20 


1 


8 


1.6 


30 


2 


7 


2-5 


40 


3 


6 


3.3 


50 


4 


6 


4.1 



5 

0.3 

0.5 
0.6 
0.7 
0-7 
1.5 
2.2 
3.0 
3.7 



P. P. 



68* 



TABLE VII.- 



33" 



-LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



157** 



' Log. Sin. d. Log. Tan. c. d. Log. Cot. Log. Cos. d. 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 

12 

13 

14 

15 

16 

17 

18 

ii 

20 

21 

22 

23 

21 

25 

26 

27 

28 

29_ 

30 

81 

32 

33 

34 



57 357 
57 389 
57 420 
57 451 
57 482 



57 513 
57 544 
57 576 
57 607 
57 638 



9-57 669 

9.57 700 

9-57 731 

57 762 

57 792 



9- 57 823 
57 854 
9. 57 885 
9-57 916 
9 57 947 



■ 57 977 

58 008 

9-58 039 

9-58 070 

-58 100 



9-58 131 
58 162 
9.58 192 
9-58 223 
9-58 253 



35 

36 

37 

38 

39_ 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49_ 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59. 

60 



58 284 
9-58 314 
58 345 
58 375 
58406 



9.58 436 

9-58 466 

9-58 497 

58 527 

58 557 



9-58 587 
9. 58 618 
9-58 648 
9-58 678 
9 ■ 58 708 



9-58 738 
9-58 769 
9-58 799 
9-58 829 
9. 58 859 



9-58 889 
9-58 919 
9-58 949 
9. 58 979 
9-59 009 



59 038 
59 068 
59 098 
59 128 
59 158 



959188 



-og. Cos. d 



31 
31 
31 
31 

31 
31 
31 
31 
31 
31 
31 
31 
31 
30 

31 
31 
31 
30 
31 
30 
31 
30 
31 
30 

30 
31 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 

30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
29 
30 
30 
29 
30 
30 



60 641 
60 677 
60 713 
60 750 
60 786 



60 822 
60 859 
60 895 
60 931 
60 967 



61 003 
61 039 
61 076 
61 112 
61 148 



61 184 
61 220 
61 256 
61 292 
61 328 



-61 364 
-61 400 
.61 436 
.61 472 
.61 507 



.61 543 
.61 579 
-61 615 
-61 651 
.61 686 



9.61 722 
9.61 758 
9-61 794 
61 829 
9-61 865 



-61 901 
.61 936 
.61 972 

• 62 007 

• 62 043 



62 078 

62 114 

9-62 149 

9.62 185 

9.62 220 



9-62 256 
9.62 291 
9.62 327 
9.62 362 
62 397 



62 433 
62 468 

9-62 503 
62 539 

9-62 574 



9-62 609 
62 644 

9-62 679 
62 715 

9-62 750 



9-62 785 

Log. Cot. c. d 



36 
36 
36 
36 

36 
36 
36 
36 
36 

36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
35 
36 
36 
35 
36 
35 
36 
35 
36 
35 
35 
36 
35 
35 
35 
35 

35 
35 
35 
35 
35 
35 
35 
35 
35 
35 

35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 



39 359 
39 322 
39 286 
39 250 
39 213 



39 177 
39 141 
39 105 
39 069 
39 032 



38 996 
38 960 
38 924 
38 888 
38 852 



38 816 
38 780 
38 744 
38 708 

38 672 



38 636 9 



38 600 
38 564 
38 528 
38 492 



■ 96 716 

• 96 711 
-96 706 
.96 701 

• 96 696 



• 96 691 
96 686 
96 681 

.96 675 

• 96 670 



96 665 
96 660 
96 655 
96 650 
96 644 



.96 639 
.96 634 
.96 629 

• 96 624 

• 96 619 



38 456 
38 420 
38 385 
38 349 
38 313 



0^38 277 
0.38 242 
0-38 206 
0.38JL70 
0-38 135 



0-38 099 
0-38 063 
0-38 028 
0-37 992 
0-37 957 



0-37 921 9 
0.37 886 



96 613 
96 608 
96 603 
96 598 
96 593 



96 587 
96 582 
96 577 
96 572 
96 567 



9.96 561 
9.96 556 
96 551 
9.96 546 
9-96 540 



9.96 535 

9^96 530 

96 525 

96 519 

96 514 



0-37 850 
0-37 815 
0-37 779 



0^37 744 
0-37 708 
0-37 673 
0-37 637 
0-37 602 



0.37 567 
0.37 531 
0.37 496 
0-37 461 
0-37 426 



0-37 390 
0-37 355 
0-37 320 
0-37 285 
0.37 250 



037 215 



Log. Tan, 



96 509 
96 503 
96 498 
96 493 
96 488 



96 482 
■ 96 477 
96 472 
96 466 
96 461 



• 96 456 
-96 450 

• 96 445 

• 96 440 

• 96 434 

• 96 429 
-96 424 
-96 418 

96 413 
96 408 



9-98 402 

Log. Sin. I d. 



45 
44 
43 
42 
41 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 



P. P. 





36 


36 


6 


3.6 


36 


7 


4.2 


4 


2 


8 


4.8 


4 


8 


9 


5-5 


5 


4 


10 


6 1 


6 





20 


12.1 


12 





30 


18.2 


18 





40 


24.3 


24 





50 


30.4 


30 








35 


35 


6 


3.5 


3.5 


7 


4 


.1 


4-1 


8 


4 


7 


4.6 


9 


5 


3 


5.2 


10 


5 


9 


5.8 


20 


11 


8 


11.6 


30 


17 


7 


17-5 


40 


23 


5 


23-3 


50 


29 


6 


29.1 



6 

7 

8 

9 

10 

20 

30 

40 

50, 



31 



s 


I 


3 


7 


4 


2 


4 


7 


5 


2 


10 


5 


15 


7 


21 





26 


2 



31 

3.1 
3.6 
4.1 

4.S 
51 
10.3 
15-5 
20.6 
25-8 



25 












^ 


24 






23 




30 30 29 


22 


6 


3.0 


3.0 


2.9 


21 


7 


3.5 


3.5 


3.4 


30 


8 


4.0 


4.0 


3-9 


19 


9 


4.6 


4.5 


4.4 


18 


10 


5.1 


5.0 


4.9 


17 


20 


10.1 


10.0 


9.8 


16 


30 


15.2 


15.0 


14.7 


15 
14 


40 


20.3 


20.0 


19-6 


50 


25.4 


25.0 


24-8 


13 






12 
11 




6 


i., 


5 
0.5 


10 




7 


0.6 


0.6 


9 




8 


0.7 


0.6 


8 




9 


0.8 


0.7 


V 




10 


0.9 


0.8 


b 




20 


1.8 


1-6 


5 




30 


2.7 


2-5 


4 




40 


3.6 


3.3 


3 




50 


4.6 


4-1 


2 






1 













' 






P 


.\ 


3^ 





113° 



627 



67' 



23° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



ise'^ 



Log. Sin, 



59 188 
59 217 
59 247 
59 277 
59 306 



59 336 
59 366 
59 395 
59 425 
59 454 



59 484 
59 513 
59 543 
59 572 
59 602 



59 631 
59 661 
59 690 
59 719 
59 749 



59 778 
59 807 
59 837 
59 866 
59 895 



59 924 
59 953 

59 982 

60 012 
60 041 



60 070 
60 099 
60 128 
60 157 
60186 



60 215 
60 244 
60 273 
60 301 
60 330 



60 359 
60 388 
60 417 
60 445 
60 474 



60 503 
60 532 
60 560 
60 589 
60 618 



60 646 
60 675 
60 705 
60 732 
60 760 



60 789 
60 817 
60 846 
60 874 
60_903 
60 93l 



d, Log. Tan. c.d. Log. Cot. Log. Cos 



Log. Cos. d. 



63 



785 
820 
855 
890 
925 
960 
995 
030 
065 
100 
135 
170 
205 
240 
275 
310 
344 
379 
414 
449 



63 484 
63 518 
63 553 
63 588 
63 622 



63 657 
63 692 
63 726 
63 761 
63 795 



63 830 
63 864 
63 899 
63 933 
63 968 



64 002 
64 037 
64 071 
64 106 
64 140 



64 174 
64 209 
64 243 
64 277 
64 312 



64 346 
64 380 
64 415 
64 449 
64 483 



64 517 
64 551 
64 585 
64 620 
64 654 



64 688 
64 722 
64 756 
64 790 
64 824 



9 64 858 
Log. Cot. 



215 
179 
144 
109 
074 
039 
004 
969 
934 
899 



36 864 
36 829 
36 794 
36 760 
36 725 



36 690 
36 655 
36 620 
36 585 
36 551 



516 
481 
447 
412 
377 
343 
308 
273 
239 
204 



36 170 
36 135 
36 101 
36 066 
36 032 



35 997 
35 963 
35 928 
35 894 
35 859 



35 825 
35 791 
35 756 
35 722 
35 688 



35 653 
35 619 
35 585 
35 551 
35 517 



482 
448 
414 
380 
346 
312 
278 
244 
209 
175 



0-35 141 
Log. Tan 



96 402 
96 397 
96 392 
96 386 
96 381 



96 375 
96 370 
96 365 
96 359 
96 354 



96 349 
96 343 
96 338 
96 332 
96 327 



96 321 
96 316 
96 311 
96 305 
96 300 



96 294 
96 289 
96 283 
96 278 
96 272 



96 267 
96 261 
96 256 
96 251 
96 245 



96 240 
96 234 
96 229 
96 223 
96 218 



96 212 
96 206 
96 201 
96 195 
96 190 



96 184 
96 179 
96 173 
96 168 
96 162 



96 157 
96 151 
96 146 
96 140 
96 134 



96 129 
96 123 
96 118 
96 112 
96 106 



96 101 
96 095 
96 090 
96 084 
96 078 



9-96 073 
log. Sin. 



60 

59 
58 
57 
51 
55 
54 
53 
52 
II 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 



P.P. 



35 



3 


5 


3. 


4 


1 


4. 


4 


7 


4. 


5 


3 


5. 


5 


9 


5. 


11 


8 


11 


17 


7 


17 


23 


6 


23. 


29 


6 


29 





34 


6 


3.4 


7 


4 





8 


4 


6 


9 


5 


2 


10 


5 


7 


20 


11 


5 


30 


17 


2 


40 


23 





50 


28 


71 



35 

5 



34 

3-4 

3.9 

4.5 

5.1 

5.6 

11.3 

17-0 

22.6 

28-8 



6 


3. 


7 


3. 


8 


4. 


9 


4. 


10 


5. 


20 


10. 


30 


15. 


40 


20. 


50 


25. 



30 


5 

5 










39 


39 


6 


2.9 


2.9 


7 


3 


4 


3 


4 


8 


3 


9 


3 


8 


9 


4 


4 


4 


3 


10 


4 


9 


4 


8 


20 


9 


8 


9 


6 


30 


14 


7 


14 


5 


40 


19 


6 


19 


3 


50 


24 


6 


24 


1 



6 

6;o.6 

7|0.7 
80.8 
9,0.9 
lOll.O 
20|2.0 
30 30 
40'4.0 
50 5.0 



5 


0.5! 





6 





7 





8 





9 


1 


8 


2 


7 


3 


g 


4 


6 



38 

2.8 

3.3 

38 

4.3 

4.7 

9.5 

14.2 

19.0 

23.7 



5 

0-5 
0.6 
0.6 
0.7 
0.8 
1.6 
2.5 
3-3 
4.1 



P. Pc 



113° 



628 



66° 



24° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



Log. Sin, 



60 931 
60 959 

60 988 

61 016 
61 044 



61 073 
61 101 
61 129 
61 157 
61 186 



61 214 
61 242 
61 270 
61 298 
61 326 



61 354 
61 382 
61410 
61438 
61 486 



61494 
61 522 
61 550 
61 578 
61 606 



61 634 
61 661 
61 689 
61 717 
61 745 



61 772 
61 800 
61 828 
61 856 
61 883 



61911 
61 938 
61 966 

61 994 

62 021 



62 049 
62 076 
62 104 
62 131 
62 158 



62 186 
62 213 
62 241 
62 268 
62 295 



62 323 
62 350 
62 377 
62 404 
62 432 



62 459 
62 486 
62 513 
62 540 
62 567 



9-62 595 



Log. Cosi 



Log. Tan. c. d. Log. Cot. Log. Cos 




64 858 
64 892 
64 926 
64 960 
64 994 



65 028 
65 062 
65 096 
65 129 
65163 



65 197 
65 231 
65 265 
65 299 
65 332 



65 366 
65 400 
65 433 
65 467 
65 501 




66 204 
66 237 
66 271 
66 304 
66 337 



66 370 
66 404 
66 437 
66 470 
66 503 



66 536 
66 570 
66 603 
66 636 
66669 



66 702 
66 735 
66 768 
66 801 
66 834 



35 141 
35 107 
35 073 
35 040 
35 006 



34 972 
34 938 
34 904 
34 870 
34 836 



34 802 
34 769 
34 735 
34 701 
34 667 



34 633 
34 600 
34 566 
34 532 
34 499 



34 465 
34 431 
34 398 
34 364 
34 331 



34 297 
34 263 
34 230 
34 196 
34 163 



34 129 
34 096 
34 062 
34 029 
3^996 



33 962 
33 929 
33 895 
33 862 
33 829 



33 795 
33 762 
33 729 
33 696 
33 662 



33 629 
33 596 
33 563 
33 529 
33 496 



9 
9 

33 463 9 
33 430 9 
33 397 
33 364 
33 331 



0-33 132 



9-66 867 

Log. Cot. c.cl.|Log. Tan. 



33 298 
33 265 
33 232 
33 198 
33 165 



96 073 
96 067 
96 062 
96 056 
96 050 



96 045 
96 039 
96 033 
96 028 
96 022 



96 016 
96 011 
96 005 
95 999 
95 994 



95 988 
95 982 
95 977 
95 971 
95 965 



95 959 
95 954 
95 948 
95 942 
95 937 



95 931 
95 925 
95 919 
95 914 
95 908 



95 902 
95 896 
95 891 
95 885 
95 879 



95 873 
95 867 
95 862 
95 856 
95 850 



95 844 
95 838 
95 833 
95 827 
95 821 



95 815 
95 809 
95 804 
95 798 
95 792 



95 786 
95 780 
95 774 
95 768 
95 763 



95 757 
95 751 
95 745 
95 739 
95 733 



9.95727 



Log. Sin. 



60 

59 
58 
57 
56 

55 
54 
53 
52 

5a 

49 
48 
47 
46 
45 
44 
43 
42 
41_ 

40 

39 
38 
37 
36 



P. P. 





34 


33 


33 


6 


3.4 


3.3 


3 3 


7 


3.9 


3.9 


3 


8 


8 


4.5 


4.4 


4 


4 


9 


5.1 


5.0 


4 


9 


10 


5.6 


5.6 


5 


5 


20 


11.3 


11.1 


11 





30 


17.0 


16.7 


16 


5 


40 


22.6 


22.3 


22 





50 


28.3 


27.9 


27 


• 5 





2^ 


6 


2.81 


7 


3 


3 


8 


3 


8 


9 


4 


3 


10 


4 


7 


20 


9 


5 


30 


14 


2 


40 


19 





50 


23 


7 



38 

2.8 

3.2 

37 

4.2 

4.6 

9.3 

14.0 

18.6 

23.3 





27 


27 


6 


2.7 


2.7 


7 


3.2 


3.1 


8 


3.6 


3.6 


9 


4.1 


4.0 


10 


4-6 


4.5 


20 


9.1 


9-0 


30 


13.7 


13.5 


40 


18.3 


18.0 


50 


22.9 


22.5 



0.6 


0. 


0.7 


0. 


0.8 





0.9 





1.0 


0. 


2.0 


1- 


3.0 


2 


4.0 


3. 


5.0 


4. 



P.P. 



114^ 



629 



TABLE VII.- 



-LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



154° 



Log. Sin 



9 



63 



595 
622 
649 
676 
703 
730 
757 
784 
811 
838 
864 
891 
918 
945 
972 
999 
025 
052 
079 
106 
132 
159 
186 
212 
239 

266 
292 
319 
345 
372 

398 
425 
451 
478 
504 
530 
557 
583 
609 
636 



662 
688 
715 
741 
767 
793 
819 
846 
872 
898 
924 
950 
976 
002 
028 
054 
080 
106 
132 
158 
184 



9-64 
Log. Cos. 



Log. Tan. c. d. Log. Cot. Log. Cos. 



66 867 
66 900 
66 933 
66 966 
66 999 



67 032 
67 065 
67 097 
67 130 
67 163 



67 196 
67 229 
67 262 
67 294 
67 327 



67 360 
67 393 
67 425 
67 458 
67 491 



67 523 
67 556 
67 589 
67 621 
67 654 



67 687 
67 719 
67 752 
67 784 
67 817 



67 849 
67 882 
67 914 
67 947 
67 979 



68 012 
68 044 
68 077 
68 109 
68 141 



68 174 
68 206 
68 238 
68 271 
68 303 



68 335 
68 368 
68 400 
68 432 
68 464 



68 497 
68 529 
68 561 
68 593 
68 625 



68 657 
68 690 
68 722 
68 754 
68 786 
68 818 
Log. Cot. 



32 
33 
33 
33 
33 
33 
32 
33 
33 

33 
32 
33 
32 
33 
32 
33 
32 
33 
32 

32 
33 
32 
32 
33 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 
32 
32 
32 
32 

32 

32 
32 
32 
32 

32 
32 
32 
32 
32 
32 
32 
32 
32 
32 

32 
32 
32 
32 
32 
32 

Z6, 



132 
100 
067 
034 
001 

968 
935 
902 
869 



32 803 
32 771 
32 738 
32 705 
32 672 



32 640 
32 607 
32 574 
32 54l 
32 509 



32 476 
32 443 
32 411 
32 378 
32 345 



32 313 
32 280 
32 248 
32 215 
32 183 



32 150 
32 118 
32 085 
32 053 
32 020 



31 988 
31 955 
31 923 
31 891 
31 858 



31 826 
31 793 
31 761 
31 729 
31 696 



31 664 
31 632 
31 600 
31 567 
31 535 



31 503 
31 471 
31 43? 
31 406 
31 374 



31 342 
31 310 
31 278 
31 246 
31 214 



31 182 



Log, Tan 



95 727 
95 721 
95 716 
95 710 
9S 704 



95 698 
95 692 
95 686 
95 680 
95 674 



95 668 
95 662 
95 656 
95 650 
95 644 



95 638 
95 632 
95 627 
95 621 
95 615 



95 609 
95 603 
95 597 
95 591 
95 585 



95 579 
95 573 
95 567 
95 561 
95 555 




95 518 
95 512 
95 506 
95 500 
95 494 



95 488 
95 482 
95 476 
95 470 
95 464 



95 458 
95 452 
95 445 
95 439 
95 433 



95 427 
95 421 
95 415 
95 409 
95 403 



95 397 
95 390 
95 384 
95 378 
95 372 
95 366 
Log. Sin, 



P. P, 



7 
8 
9 
10 
20 
30 
40 
50 



33 

3 
3 

4 

4 

5 
11 
16 
22 
27 



33_ 

3-2 



33 

3.2 

3-7 

4.2 

4.8 

5.3 

10.6 

16.0 

21-3 

26.6 



10 
20 
30 
40 
50 



27 

2.7 





26 


26 


2 


6 


2.6 


2-6 


2. 


7 


3.1 


3 





3. 


8 


3.5 


3 


4 


3 


9 


4.0 


3 


9 


3 


10 


4.4 


4 


3 


4. 


20 


8.8 


8 


6 


8. 


30 


13.2 


13 





12. 


40 


17.6 


17 


3 


17. 


50 


22.1 


21 


6 


21. 





6 


6 


S 


6 


0.6 


0.6 





7 


0.7 


0.7 





8 


0.8 


0.8 





9 


1.0 


0.9 





10 


1.1 


1.0 





20 


2.1 


2.0 


1 


30 


3-2 


3.0 


2 


40 


4.3 


4.0 


3 


50 


5.4 


5.0 


4. 



P. p. 



630 



64- 



TABLE VII.— LOGARITHMIC SINES. COSINES. TANGENTS, 
AND COTANGENTS. 



' Log. Sin. d. Log. Tan. c. d, Log. Cot. Log. Cos. 

77 Q.RA 184. TT Q.RR 818 TT 0.31 182 9-95 366 



153^ 



64 184 
64 210 
64 236 
64 262 
64 287 



64 313 
64 339 
64 365 
64 391 
64 416 



64 442 
64 468 
64 493 
64 519 
64 545 



64 570 
64 596 
64 622 
64 647 
64 673 



64 698 
64 724 
64 749 
64 775 
64 800 



64 826 
64 851 
64 876 
64 902 
64 927 



64 952 

64 978 

65 003 
65 028 
65 054 



65 079 
65 104 
65 129 
65 155 
65 180 



65 205 
65 230 
65 255 
65 280 
65 305 



65 331 
65 356 
65 381 
65 406 
65 431 



65 456 
65 481 
65 506 
65 530 
65 555 



65 580 
65 605 
65 630 
65 655 
65 680 



9-65 704 
Log, Cos 



.68 818 

• 68 850 

• 68 882 

• 68 914 
9-68 946 



9.69 615 
9.69 647 
9.69 678 
9.69 710 
9.69 742 



9.68 978 

69 010 

69 042 

9-69 074 

9-69 108 



9- 69 138 
9.69 170 
9-69 202 
9.69 234 
9-69 265 



9-69 297 
9.69 329 
9.69 361 
9.69 393 
69 425 



69 456 

69 488 

9.69 520 

9.69 552 

9.69 583 



9.69 773 
9-69 805 
9.69 837 
9.69 868 
9-69 900 



9-69 931 
9-69 963 

9.69 994 

9.70 026 
9.70 058 



9.70 089 
9.70 121 
9.70 152 
9.70 183 
9.70 215 



9.70 246 
9.70 278 
9.70 309 
9.70 341 
9.70 372 



9.70 403 
9.70 435 
9.70 466 
9.70 497 
9.70 529 



9.70 560 
70 591 

9.70 623 
70 654 
70 685 



9.70 716 
Log. Cot, 



0.31 
0.31 
0.31 
0.31 
0.31 



0.31 
0.30 
0.30 
0.30 
0.30 



182 
150 
117 
085 
053 
02l 
989 
957 
926 
894 



0.30 
0.30 
0.30 
0.30 
0.30 



862 
830 
798 
766 

734 



0.30 
0-30 
0.30 
0.30 
0.30 



702 
670 
639 
607 
575 



0.30 
0.30 
0.30 
0.30 
Q.30 



543 
511 
480 
448 

416 



0.30 
0.30 
0.30 
0.30 
0.30 



384 
353 
321 
289 
258 



0.30 
0.30 
0.30 
0-30 
0.30 



226 
194 
163 
131 
100 



0.30 
0.30 
0.30 
0.29 
0.29 



9 

0379 

005 

973 

942 



0.29 
0.29 
0.29 
0.29 
0.29 



0.29 
0.29 
0.29 
0.29 
0.29 



0.29 
0.29 
0.29 
0.29 
0.29 



910 
879 
847 



785 
753 
722 
690 
659 
628 
596 
565 
533 
502 
471 



0.29 
0.29 
0.29 
0.29 
0.29 



439 
408 
377 
346 
314 



0.29 283 



Log. Tan, 



95 366 
95 360 
95 353 
95 347 
95 341 



95 335 
95 329 
95 323 
95 316 
95 310 



95 304 
95 298 
95 292 
95 285 
95 279 



95 273 
95 267 
95 260 
95 254 
95 248 



95 242 
95 235 
95 229 
95 223 
95217 



95 210 
95 204 
95 198 
95 191 
95 185 



95 179 
95 173 
95 166 
95 160 
95 154 



95 147 
95 141 
95 135 
95 128 
95 122 



95 116 
95 109 
95 103 
95 097 
95 090 



95 084 
95 078 
95 07l 
95 065 
95 058 



95 052 
95 046 
95 039 
95 033 
95 026 



95 020 
95 014 
95 007 
95 001 

94 994 



9-94 988 
Log. Sin, 



P.P. 



3»_ 

3.2 
3.8 

4.3 
4.9 
5.4 
10.8 
16-2 
21-6 
,27.1 





31 


6 


3.1 


7 


3.7 


8 


4.2 


9 


4.7 


10 


5.2 


20 


10.5 


30 


15.7 


40 


21.0 


50 


26.2 



33 

3.2 
3.7 

4.2 
4.8 

5.3 
10.6 
16.0 
21.3 
26.6 



31 

3.1 

3.6 
4.1 

4-6 
5.1 

10.3 
15.5 
20.6 
25. § 





36 


35 


6 


2.6 


2.5 


7 


3.0 


3.0 


8 


3.4 


3.4 


9 


3-9 


3-8 


10 


4-3 


4-2 


20 


8.6 


8-5 


30 


13-0 


12-7 


40 


17.3 


17-0 


50 


21.6 


21.2 





31 


6 




6 


2.4 


0.61 


7 


2.8 





7 


8 


3.2 





8 


9 


3.7 


1 





10 


4.1 


1 


1 


20 


8.1 


2 


1 


30 


12.2 


3 


2 


40 


16-3 


4 


3 


50 


20.4 


5 


4 



35 

2.L 

2.9 

3.3 

3-7 

4-1 

8.3 

12.5 

16.6 

20. S 



6 

0.6 
0-7 
0-8 
0-9 
1.0 
2.0 

4.0 
5.0 



P.P. 



631 



63*' 



37* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



153* 



Log. Sin. 



o 

1 

2 
3 

5 

6 

7 

8 

-9. 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 



9.65 
9.65 
9.65 
9.65 
9.65 

9.65 
9-65 
9.65 
9.65 
9.65 



9.65 
9.65 
9.66 
9.66 
9.66 

9.66 
9.66 
9.66 
9.66 
9.66 



30 

21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 
S3 
34 
35 
36 
37 
38 
39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 



9.66 
9.66 
9.66 
9.66 
9.66 



9.66 
9.66 
9.66 
9.66 
9.66 



9.66 
9.66 
9.66 
9.66 
9.66 



9.66 
9.66 
9.66 
9.66 
9.66 



9.66 
9.66 
9.66 
9.66 
9.66 



9.66 
9.66 
9.66 
9.66 
9.66 
9.66 
9.66 
9.66 
66 
9-67 
9-67 
9.67 
9.67 
9.67 
9-67 
9.67 



704 
729 
754 
779 
803 
828 
853 
878 
902 
927 
95l 
976 
001 
025 
050 
074 
099 
123 
148 
172 
197 
22l 
246 
270 
294 
319 
343 
367 
392 
416 
440 
465 
489 
513 
537 
56l 
586 
610 
634 
658 
682 
706 
730 
754 
778 
802 
826 
850 
874 
898 
922 
946 
970 
994 
018 
042 
066 
089 
113 
137 
161 



Log. Cos. 



Log. Tan, 



9.70 716 
70 748 

9.70 779 
70 810 
70 841 



70 872 
9.70 903 
9.70 935 
9.70 966 
9.70 997 



9.71 028 
9.71059 
9.71090 
9.71 121 
9.71 152 



9.71 183 
9.71 214 
9.71 245 
9.71 276 
9.71 307 



9.71 338 
9.71369 
9.71400 
9.71431 
9.71462 



9.71493 
9.71 524 
9.71 555 
9.71 586 
9.71 617 



9.71 647 
9.71 678 
9.71 709 
9.71740 
9.71 771 



9.71 801 
9.71 832 
9.71 863 
9.71894 
9.71 925 



9.71 955 

9.71 986 

9.72 017 
9.72 047 
9.72 078 



9.72 109 
9.72 139 
9.72 170 
9.72 201 
9.72 231 



9.72 262 
9.72 292 
9.72 323 
72 354 
9.72 384 



9.72 415 
9 . 72 445 
9.72 476 
9.72 506 
9.72 537 



9-72 587 



C.d 



Log. Cot- 



Log. Cot Log. Cos 



0.29 283 
0.29 252 
0.29 221 
0.29 190 
0.29 158 



0.29 127 
0.29 096 
0.29 065 
0.29 034 
0.29 003 



0.28 972 
0.28 940 
0.28 909 
0.28 878 
0.28 847 



0.28 816 
0.28 785 
0.28 754 
0.28 723 
0.28 692 



0.28 661 
0.28 630 
0.28 599 
0.28 568 
0.28 537 



0.28 506 
0.28 476 
0.28 445 
0.28 414 
0.28 383 



0.28 
0.28 
0.28 
0.28 
0.28 



0.28 
0.28 
0.28 
0.28 
0.28 



352 
321 
290 
260 
229 
198 
167 
136 
106 
075 



0.28 044 
0.28 014 
0.27 983 
0.27 952 
0.27 921 



0.27 891 
0.27 860 
0.27 830 
0.27 799 
0.27 768 



0.27 738 
0.27 707 
0.27 677 
0.27 646 
0-27 615 




.94 988 
.94 981 
.94 975 
.94 969 
.94 962 



.94 956 
. 94 949 
. 94 943 
.94 936 
.94 930 



94 923 
94 917 
94 910 
94 904 
94 897 



.94 891 
.94 884 
.94 878 
.94 871 
.94 865 



9. 



94 858 
94 852 
94 845 
94 839 
94 832 



.94 825 
.94 819 
.94 812 
. 94 806 
.94 799 



94 793 
94 786 
94 779 
94 773 
94 766 



94 760 
94 753 
94 746 
94 740 
94 733 



9.94 727 
94 720 
94 713 
94 707 

9.94 700 



9.94 693 
9.94 687 
9.94 680 
9.94 674 
9.94 667 



9.94 660 
9.94 654 
9.94 647 
9 . 94 640 
9.94 633 



9.94 627 
9.94 620 
9.94 613 
9.94 607 
9.94 600 



9-94 593 



117^ 



c.d. [Log. Tan. Log. Sin 
632 



P.P. 





31 


31 


6 


3.1 


3.1 


7 


3.7 


3.6 


8 


4.2 


4.1 


9 


4.7 


4.6 


10 


5.2 


5.1 


20 


10.5 


10.3 


30 


15.7 


15.5 


40 


21.0 


20.6 


50 


26.2 


25.81 



30 

3.0 

3.5 

4.0 

4.6 

5.1 

10. 1 

15.2 

20.3 

25.4 



6 
7 
8 
9 

10 
20 
30 
40 
50 



35 

2.5 

2.9 

3.3 

3.7 

4.1 

8.3 

12.5 

16.6 

20.8 





3? 


34 


31 


6 


2.4 


2.4 


2.3 


7 


2.8 


2.8 


2.7 


8 


3.2 


3.2 


3.1 


9 


3.7 


3.6 


3.5 


10 


4.1 


4.0 


3.9 


20 


8.1 


8.0 


7.8 


30 


12.2 


12.0 


11.7 


40 


16.3 


16.0 


15.6 


50 


20.4 


20.0 


19.6 





7 


6 


€ 




6 


0.7 


0.6 


0.6 


7 





8 


0.7 





7 


8 





9 


0.8 





8 


9 


1 





1.0 





9 


10 


1 


1 


1.1 


1 





20 


2 


3 


2.1 


2 





3013 


5 


3.2 


3 





4014 


6 


4-3 


4 





50 


5 


8 


5.4 


5 






P.P. 



63* 



38* 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



151° 



Log. Sin. d. 



9.67 161 
9.67 184 
9.67 208 
9.67 232 
9.67 256 



9.67 279 
9.67 303 
9.67 327 
9.67 350 
67374 



9.67 397 
9.67 421 
9.67 445 
9.67 468 
9.67 492 



9.67 515 
9.67 539 
9.67 562 
9.67 586 
9-67 609 



9.67 633 
9.67 656 
9.67 679 
9.67 703 
9-67 726 



9.67 750 
67 773 

9.67 796 
67 819 

9.67 843 



9.67 866 
9.67 889 
9.67 913 
9.67 936 
9.67 959 



9.67 982 

9.68 005 
9.68 029 
9.68 052 
9-68 075 



9-68 098 
9.68 121 
9.68 144 
9.68 167 
9.68 190 



9.68 213 
9.68 236 
.68 259 
9.68 282 
9.68 305 

9.68 328 
9.68 351 
9.68 374 
9.68 397 
9.63 420 
9.68 443 
9.68 466 
9.68 488 
9.68 511 
9.68 534 



9-68 557 



Log. Cos. 



Log. Tan. led. I Log. Cot. Log. Cos 



72 567 
72 598 
72 628 
72 659 
72689 



72 719 
72 750 
72 780 
72 811 
72 841 



72 871 
72 902 
72 932 
72 962 
72 993 



73 023 
73 053 
73 084 
73 114 
73 144 



73 174 
73 205 
73 235 
73 265 
73 295 



73 325 
73 356 
73 386 
73 416 
73 446 



73 476 
73 506 
73 536 
73 567 
73 597 




73 927 
73 957 

73 987 

74 017 
74 047 



74 076 
74 106 
74 136 
74 166 
74 196 



74 226 
74 256 
74 286 
74 315 
74 345 



9-74 375 



Log. Cot. 



30 
30 
30 
30 
30 
30 
30 
30 
30 

30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 

30 
30 
30 
30 
30 
30 
30 
30 
30 
30 

30 
30 
30 
30 
30 

29 
30 
30 
30 
29 

30 
30 
30 
29 
30 
29 



27 432 
0.27 402 
0.27 371 
0.27 341 
0.27 311 



0.27 280 9 
0.27 250 9 
0.27 219 
0.27 189 
0.27 159 



0.27 128 
0.27 098 
0.27 067 
0.27 037 
0.27 007 



9. 



94 593 
94 587 
94 580 
94 573 
94 566 



0.26 976 
0.26 946 
0.26 916 
0.26 886 
0.26 855 



0.26 825 
0.26 795 
0.26 765 
0.26 734 
0^26 704 



0.26 
0.26 
0.26 
0.26 
0.26 



674 
644 
614 
584 
553 



0.26 
0.26 
0.26 
0.26 
0.26 



523 
493 
463 
433 
403 



0.26 373 
0.26 343 
0.26 313 
0.26 283 
0-26 253 



0.26 22S 
0.26 193 
0.26 163 
0.26 133 
0.26 103 



0.26 073 
0.26 043 
0.26 013 
0.25 983 
0_;_25_953 
0.25 923 
0.25 893 
0.25 863 
0.25 833 
0.25 804 



0.25 774 
. 25 744 
0.25 714 
0.26 684 
0-25 654 
0.25 625 



.94 560 
. 94 553 
. 94 546 
.94 539 
.94 533 



9.94 526 
9.94 519 
9.94 512 
9.94 506 
9.94 4 99 



9.94 492 
9.94 485 
9.94 478 
9.94 472 
9-94 465 



9.94 458 
9 . 94 45l 
9.94 444 
9.94 437 
9.94 431 



. 94 424 
94 417 
94 410 

. 94 403 
94 396 



94 390 
94 383 
94 376 
94 369 
94 362 



94 355 
94 348 
94 341 
94 335 
94 328 



.94 321 
.94 314 
.94 307 
. 94 300 
• 94293 



94 286 
94 279 
94 272 
94 265 
94 258 



.94 251 

• 94 245 
.94 238 
.94 231 

• 94 224 



94 217 
9.94 210 
9.94 203 
9.94 196 
9.94 189 
9.94 182 



118** 



Log. Tan. Log. Sin. 
633 



60 

59 
58 
57 
5i 
55 
54 
53 
52 
51 

50 

49 
48 
47 

45 
44 
43 
42 

40 

39 
38 
37 
36 
35 
34 
33 
32 
11 
30 
29 
28 
27 
26 
25 
24 
23 
22 

21. 
20 
19 
18 
17 
16. 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6_ 
5 
4 
3 
2 

_1 

O 



P. P. 





30 


no 


6 


3.0 


$.0 


7 


3.5 


3.5 


8 


4.0 


4.0 


9 


4.6 


4.5 


10 


5.1 


5.0 


20 


10.1 


10.0 


30 


15.2 


15.0 


40 


20.3 


20.0 


50 


25.4 


25.0 



29_ 

2.9 

3.4 

3.9 

4.4 

4.9 

9.8 

14.7 

19.6 

24.6 





24 


6 


2.4 


7 


2.8 


8 


3.2 


9 


3.6 


10 


40 


20 


8.0 


30 


12.0 


40 


16.0 


50 


20.0 





23 


33 


6 


2.3 


2.3 


7 


2.7 


2.7 


8 


3.1 


3^0 


9 


3.5 


3^4 


10 


3.9 


3^8 


20 


7.8 


7.6 


30 


11.7 


11.5 


40 


15.6 


15.3 


50 


rg.e 


19.1 



22 

2.2 

2.6 

3.0 

3.4 

3.7 

7.5 

11.2 

15.0 

18. 7 





7 




6 


0.71 


7 





8 


8 





9 


9 


1 





10 


1 


1 


20 


2 


3 


30 


3 


5 


40 


4 


6 


50 


5 


8 



6 

0.6 
0.7 
0.8 
1.0 
1.1 
2-I 
3.2 
4.3 
5.4 



P.P. 



61* 



»9 



TABLE VIL—LOGARITHMIC SINES, COSINES, TANGENTS, 

AND COTANGENTS. 



ISO* 



5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 



30 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Log. Sin. 



68 557 
68 580 
68 602 
68 625 
68 648 



68 671 
68 693 
68 716 
68 739 
68761 



68 784 
68 807 
68 829 
68 852 
68 874 



68 897 

68 920 
68 942 
68 965 
68 987 



69 010 
69 032 
69 055 
69 077 
69 099 



69 122 
69 144 
69 167 
69 189 
69 211 



• 69 234 

• 69 256 

• 69 278 

• 69 301 

• 69 323 



69 345 
69 367 
69 390 

69 412 
69 434 



9.69 456 
9.69 478 
9-69 500 
9-69 523 
9^69 545 

9^69 567 
9-69 589 
9^69 611 
9-69 633 
9.69 655 



9.69 677 
9^69 699 
9^69 721 
9-69 743 
9.69 765 



9.69 787 
9.69 809 
9.69 831 
9.69 853 
9.69 875 



9.69 897 



Log, Cos. 



9.74 524 
9.74 554 
9.74 583 
9.74 613 
9.74 643 



9.74 672 
9.74 702 
9.74 732 
9.74 761 
9.74 791 



9.74 821 
9.74 850 
9.74 880 
9.74 909 
9.74 939 



Log. Tan. c, d» Log. Cot. Log. Cos. 



9-74 370 
9-74 405 
9-74 435 
9-74 464 
9-74 494 



9.74 969 

9.74 998 

9.75 028 
9.75 057 
9.75 087 



9.75 116 
9.75 146 
9.75 175 
9.75 205 
9-75 234 



9.75 264 
9.75 293 
9.75 323 
9.75 352 
9.75 382 



9.75 411 
9.75 441 
9.75 470 
9-75 499 
9-75 529 



9.75 558 
9.75 588 
9.75 617 
9.75 646 
9-75 676 



9.75 705 
9.75 734 
9-75 764 
9-75 793 
9-75 822 



9-75 851 
9-75 881 
9-75 910 
9-75 939 
9-75 9 

9-75 998 
9-76 027 
9-76 056 
9-76 085 
76 115 



9^76 144 
Log. Cot. 



0-25 625 
0-25 595 
0.25 565 
0-25 535 
0^25 505 
0-25 476 
0-25 446 
0-25 416 
0-25 387 
0.25 357 



9.94 182 
9.94175 
9.94 168 
9.94161 
9.94 154 



9.94 147 
94 140 
9 •94 133 
9^94 126 
9.94 118 



0-25 327 
0-25 297 
0-25 268 
0-25 238 
0-25 208 



0-25 179 
0-25 149 
0-25 120 
0-25 090 
0-25 060 



0-25 031 
0.25 001 
0.24 972 
0-24 942 
0-24 913 



0.24 883 
0.24 854 
0-24 824 
0-24 795 
0.24 765 



0-24 736 
0-24 706 
0-24 677 
0-24 647 
0jl24_618 
0-24 588 
0-24 559 
0.24 529 
0-24 500 
0.24 471 



. 24 441 
0.24 412 
0.24 383 
0.24 353 
0.24 324 



0.24 295 
0.24 265 
0.24 236 
0.24 207 
0.24 177 



0.24 148 
0.24 119 
0.24 090 
0.24 060 
0.24 031 



0.24 002 
0.23 973 
0.23 943 
0.23 914 
0.23 885 



0.23 856 
Log, Tan, 



9.94 111 
9.94 104 
9.94 097 
9.94 090 
9.94 083 



9-94 076 
9-94 069 
9-94 062 
9.94 055 
9 ■ 94 048 




9.93 934 
9.93 926 
9.93 919 
9.93 912 
9.93 905 



9.93 898 
9.93 891 
9.93 883 
9.93 876 
9.93 869 



9.93 862 
9.93 854 
93 847 
9.93 840 
9.93 883 



9.93 826 
9.93 818 
9.93 8ll 
9.93 804 
9.93 796 



9.93 789 
9.93 782 
9.93 775 
9-93 767 
93 760 



9- 93 753 
Log, Sin 



60 

59 
58 
57 
56^ 

55 
54 
53 
52 
51 
50 
49 
48 
47 

45 
44 
43 
42 
41 

40 

39 
38 

37 
M. 
35 
34 
33 
32 
11 
30 
29 
28 
27 
26 

25 
24 
23 
22 
21 



P. P. 





30 


29„ 


2i 


6 


3.0 


2.9 


2 


7 


8.5 


3 


4 


3 


8 


4.0 


3 


9 


3 


9 


45 


4 


4 


4 


10 


5.0 


4 


9 


4 


20 


10.0 


9 


8 


9. 


30 


15.0 


14 


7 


14. 


40 


20.0 


19 


6 


19. 


50 


25-0 


24 


6 


24. 



10 
20 
30 
40 
50 



23 

2.3 

2-7 

3.0 

34 

3.8 

7.6 

11.5 

15.3 

19.1 





7 


6 


0.7 


7 


0.9 


8 


1.0 


9 


1.1 


10 


1.2 


20 


2.5 


30 


3-7 


40 


5.0 


50 


6.2 



0.7 
0-8 
0.9 
1.5 
1.1 
2.3 
3.5 
4-6 
5.8 



P.P. 





23 


33 


21 


6 


2.2 


2.2 


2.1 


7 


2.6 


2.5 


2 


5 


8 


3.0 


2.9 


2 


3 


9 


3.4 


3-3 


3 


2 


10 


3.7 


3-6 


3 


Q 


20 


7.5 


7-3 


7 


1 


30 


11.2 


11.0 


10 


7 


40 


15.0 


14.6 


14 


3 


50 


18.7 


18-3 


17 


9 



119^ 



634 



60' 



30^^ 



TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS, 
AND COTANGENTS. 



149* 



O 

1 
2 
S 

5 

6 

7 

8 
A 
10 
11 
12 
13 
14 



9. 69 
9.69 
9.69 
69 
9-69 



9.70 

9.70 

70 

9.70 

9.70 
9.70 
9.70 
9.70 
9.70 



15 
16 
17 
18 
19 



20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



Log. Sin. 



9.70 
9.70 
9.70 
9.70 
9.70 



897 
919 
940 
962 
981 
006 
028 
050 
071 
093 
115 
137 
158 
180 
202 
223 
245 
267 
288 
310 



9.70 
9. 70 
9.70 
9.70 
9.70 



831 
353 
375 



9.70 
9-70 
9.70 
9.70 
9-70 



9. 70 
9. 70 
9.70 
9.70 
9. 70 



I 40 

I 41 

42 

I 43 

I 44 

I 45 

I 46 

( 47 

I 48 

I 49 



9. 70 

9.70 

.70 

.70 

70 



60 

51 
52 
53 
54 

55 
56 
57 
58 
59 



60 



418 
439 
461 
482 
504 
525 

547 
568 
590 
611 
632 

654 
675 
696 
718 
739 



Log. Tan. c.d. Log. Cot. Log, Cos. 



22 
21 
22 
22 
21 
22 
22 
21 
22 
2l 
22 
21 
21 
22 

21 
21 
22 
21 
21 

2l 
22 
21 
21 
21 

2l 
21 
21 
21 
21 

21 
21 
2l 
21 
^21 
21 
2l 
21 
2l 



9.76 144 
9.76 173 
9.76 202 
9-76 231 
9.76 260 



9-70 
9.70 
9.70 
9.70 
9. 70 



9.70 
9.70 
9.71 
9.71 
9-71 



760 
782 
803 
824 
846 
867 
888 
909 
930 
952 

973 
994 
015 
036 
057 



9.71 
9.71 
9.71 
9.71 
9.71 



078 
099 
121 
142 
163 



21 

21 
21 
21 
2l 
2l 
21 
21 
2l 
21 
21 

21 
21 
2l 
21 
21 
21 
21 
21 
21 
121 



9-76 289 
9.76 319 
9.76 348 
9.76 377 
76 406 



9-76 435 
9-76 464 
9-76 493 
9-76 522 
9-76 551 



g-71 184 



Log. Cos. d, 



21 



9-76 580 
9-76 609 
9-76 638 
9-76 667 
9.76 696 



9-76 725 
9-76 754 
9-76 783 
76 812 
9-76 841 



9-76 870 
9-76 899 
9-76 928 
9.76 957 
9.76 986 



9.77 015 
9.77 043 
9.77 072 
9.77 101 
9.77 130 



77 159 

77 188 

9.77 217 

9.77 245 

9-77 274 



9.77 303 
9.77 332 
9.77 361 
9.77 389 
9-77 418 



9-77 447 
9-77 476 
9-77 504 
9-77 533 
9-77 562 



9.77 591 
9-77 619 
9-77 648 
9-77 677 
9-77 705 



9-77 734 
77 763 
9-77 791 
9-77 820 
9-77 849 



9.77 877 
Log. Cot. c. d 



0-23 856 
0.23 827 
0.23 797 
0.23 768 
0.23 739 



0.23 
0.23 
0.23 
0.23 
0.23 



0.23 
0.23 
0.23 
0.23 
0.23 



710 
681 
652 
623 
594 

565 
535 

506 
477 
448 



0.23 419 
0.23 390 
0.23 361 
0-23 332 
0-23 303 



0.23 274 
0-23 245 
0-23 216 
0-23 187 



93 753 
93 746 
93 738 
93 731 
93 724 



93 716 
93 709 
.93 702 
.93 694 
93 687 



-93 680 
-93 672 
-93 665 
-93 658 
.93 650 



- 93 643 
-93 635 
-93 628 
-93 621 
•93 613 



9-93 606 

9.93 599 

9.93 591 

93 584 



0.23 158 9.93 576 



0-23 129 
0-23 101 
0.23 072 
0.23 043 
0-23 014 



0-22 985 
0-22 956 
0-22 927 
0-22 898 
0-22 &69 



0-22 841 
0-22 812 
0-22 783 
0-22 754 
0.22 725 



0-22 696 
0-22 668 
0-22 639 
0-22 610 
0-22 581 



0-22 553 
0.22 524 
0-22 495 
0.22 466 
0-22 438 



0-22 409 
0-22 380 
0.22 352 
0.22 323 
0.22 294 




0-22 122 



Log. Tan 



93 569 

93 562 

93 554 

9.93 547 

9.93 539 




9-93 457 
9-93 450 
93 442 
9.93 435 
9-93 427 



9-93 420 
9-93 412 
9-93 405 
9-93 397 
9.93 390 




9.93 306 
Log. Sin. 



60 

59 
58 
57 
11 
55 
54 
53 
52 
51 

50 

49 
48 

47 
46 

45 
44 
43 
42 
41 

40 

39 
38 
37 
11 
35 
34 
33 
32 
31 

30 

29 
28 
27 
26 
25 
24 
23 
22 
21 

20 
19 
18 
17 

JA 
15 
14 
13 
12 
11 

10 



P.P. 





29 


29 


6 


2.9 


2-9 


7 


3-4 


3-4 


8 


3-9 


3-8 


9 


4-4 


4-3 


10 


4.9 


4-8 


20 


9-8 


9-6 


30 


14.7 


14.5 


40 


19.6 


19.3 


50 


24.6 


24.1 



28. 

2.8 

3-3 
3.8 
4.3 
4.7 
9.5 
14.2 
19.0 
7 





22 


21 


6 


2-2 


2.1 


7 


2-5 


2-5 


8 


2-9 


2-8 


9 


3.3 


3-2 


10 


3.6 


3-6 


20 


7.3 


7-1 


30 


11.0 


10.7 


40 


14.6 


14.3 


50 


18.3 


17.9 



21 

2.1 
2-3 

i:f 

3.5 

70 

10.5 

14.0 

17-5 



8 7_ 



0.8 


0-7 


0-9 


0-9 


1-0 


1-0 


1-2 


1-1 


1-3 


1-2 


2-6 


2-5 


4-0 


3-7 


5-3 


5-0 


6.6 


6.2 



7 
0.7 
0.8 
0.9 
1.0 
l.I 
2-3 
3.5 
4.6 
5.8 



P.P. 



ISO?^ 



635 



59* 



TABLE Vir.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



148° 



Log. Sin. 




71 184 
71 205 
71226 
71247 
71268 



71289 
71310 
71331 
71351 
71 372 



71393 
71414 
71435 
71456 
71477 



71 705 
71 726 
71746 
71767 
71 788 



71808 
71829 
71 849 
71870 
71 891 



71 911 
71 932 
71 952 
71973 
71 993 



72 014 
72 034 
72 055 
72 075 
72 096 



72 116 
72 136 
72 157 
72 177 
72 198 



72 218 
72 238 
72 259 
72 279 
72 299 



72 319 
72 340 
72 360 
72 380 
72 400 
72 421 
Log. Cos. 



9.78 590 
78 618 
9.78 647 
9.78 675 
9-78 703 



Log. Tan. c.d. Log. Cot. Log. Cos 



9.77 877 
9-77 906 
77 934 
9.77 963 
9.77 992 



9.78 020 
9-78 049 
9.78 077 
9.78 106 
9.78 134 



9.78 163 
9-78 191 
9.78 220 
9-78 248 
9.78 277 



9.78 305 
9.78 334 
9.78 362 
9.78 391 
9-78 419 



9.78 448 
9.78 476 
9.78 505 
9.78 533 
9.78 561 



9.78 732 
9.78 760 
9.78 788 
9.78 817 
9.78 845 



9.78 873 
9.78 902 
9.78 930 
9.78 958 
9.78 987 



9.79 015 
9.79 043 
9.79 071 
9.79 100 
9.79 128 



9.79 156 
9-79 184 
9.79 213 
9.79 241 
9.79 269 



9.79 297 

9.79 325 

79 354 

79 382 

9. 79 410 



9-79 438 
9.79 466 
9-79 494 
9-79 522 
9^7J_551 
9 • 79 579 
Log. Cot, 



0-22 122 
0-22 094 
0-22 065 
0.22 037 
0.22008 



0.21 410 
0.21 381 
0.21 353 
0.21 325 
0.21 296 



0.21 
0.21 
0.21 
0.21 
0.21 



0.21 
0.21 
0.21 
0.21 
0.21 



979 
951 
922 
894 
865 
837 
808 
780 
751 
723 



0.21 
0.21 
0.21 
0.21 
0.21 



694 
666 
637 
609 
580 



0.21 
0.21 
0.21 
0.21 
0.21 



552 
523 
495 
467 
438 



0.21 268 
0.21 239 
0.21 2ll 
0.21 183 
0.21 154 



0.21 126 
0.21 098 
0.21 070 
0.21 041 
0.21 013 



0.20 985 
0-20 956 
0.20 928 
0-20 900 
0-20 872 



0-20 843 
0-20 815 
0-20 787 
0.20 759 
0-20 731 



0.20 702 
0.20 674 
0-20 646 
0-20 618 
0-20 590 



0-20 561 
0-20 533 
0-20 505 
0-20 477 
Qj_20_449 
0-20 421 
Log, Tan 



93 306 
93 299 
93 291 
93 284 
93 276 
93 268 
93 261 
93 253 
93 245 
93 238 



93 230 
93 223 
93 215 
93 207 
93 200 



93 192 
93 184 
93 177 
93 169 
93 161 



93 153 
93 146 
93 138 
93 130 
93 123 



93 115 
93 107 
93 100 
93 092 
93 084 



93 076 
93 069 
93 061 
93 053 
93 045 



93 038 
93 030 
93 022 
93 014 
93 006 



92 999 
92 991 
92 983 
92 975 
92 967 



92 960 
92 952 
92 944 
92 936 
92 928 



92 920 
92 913 
92 905 
92 897 
92 889 



92 881 
92 873 
92 865 
92 858 
92 850 



131** 



9-92 842 
Log. Sin. 

636 



P. P. 





29 


28 


28 


6 


2.9 


2.8 


2.8 


7 


3 


4 


3 


3 


3 


2 


8 


3 


8 


3 


8 


3 


7 


9 


4 


3 


4 


3 


4 


2 


10 


4 


8 


4 


7 


4 


g 


20 


9 


6 


9 


5 


9 


3 


30 


14 


5 


14 


2 


14 





40 


19 


3 


19 





18 


3 


50 


24 


1 


23 


7 


23 


^ 



21 

2.1 

2-4 

2.8 

3-1 

3.5 

7.0 

10.5 

14.0 

17.5 



20 

2.0 

2.4 

2-7 

3.1 

3.4 

6-8 

10.2 

13.6 

17-1 



20 

2.0 

2.3 

2.6 

3.0 

3.3 

6.6 

10.0 

13.3 

16.6 



8 ^« 

0.7 

0.9 
10 

1.1 

1.2 
2.5 
7 
50 
6.2 






8 





9 


1 





1 


2 


1 


3 


2 


6 


4 





5 


3 


6 


6 



P. p. 



68** 



as'* 



TABLE VII.— LOGARITHMIC SINES, COSINES. TANGENTS. 

AND COTANGENTS. 147* 



O 

1 

2 

3 
J_ 

5 

6 

7 

8 
_9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 



30 

21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 

33 9 

34 9 

35 
36 
37 
38 
39_ 
40 
41 
42 
43 
44 



Log. Sin. 



72 421 
72 441 
72 461 
72 481 
72 50l 



72 522 
72 542 
72 562 
72 582 
72 602 



72 622 
72 642 
72 662 
72 682 
72 702 



72 723 
72 743 
72 763 
72 783 
72 802 



.72 822 
.72 842 
.72 862 
• 72 882 
.72 902 



72 922 
72 942 

72 962 
.72 982 

73 002 



9.73 120 
9.73 140 
9.73 160 
9.73 180 
9-73 199 



45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



73 021 
73 041 
73 061 
73 081 
73 101 



Log. Tan. c. d.JLog. Cot. Log. Cos 



9.73 219 
9.73 239 
9.73 258 
9.73 278 
9.73 298 



9.73 317 
9.73 337 
9.73 357 
73 376 
9-73 396 



9.73 415 

9.73 435 

9.73 455 

73 474 

73 494 



73 513 
73 533 
73 552 
73 572 
73 591 



9.73 611 



20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
20 
19 
20 
20 
20 
20 
20 
20 
19 
20 
20 
20 

19 
20 
20 
19 
20 

19 
20 
19 
20 
19 
20 
19 
19 
20 
19 
19 
20 
19 
19 
19 
19 
20 
19 
19 
19 

19 
19 
19 
19 
19 
19 



79 579 
9-79 607 
9.79 635 
9.79 663 
9.79 691 



Log. Cos, 



79 719 
9.79 747 
9.79 775 
9.79 803 

79 831 



9.79 859 
9.79 887 
79 915 
9.79 943 
9.79 971 



79 999 
9.80 027 

80 055 
9.80 083 

80 111 



9.80 139 
9.80 167 
9.80 195 
9.80 223 
9.80 251 



9-80 279 
9.80 307 
9.80 335 
9.80 363 
9-80 391 



9.80 418 
9 . 80 446 
9 . 80 474 
9.80 502 
9.80 530 



9.80 558 
80 586 
9.80 618 
9.80 641 
9.80 669 



9.80 697 
9.80 725 
9.80 752 
9.80 780 
9.80 808 



0.20 280 
0.20 252 
0.20 224 
0-20 196 
0.20 168 



80 836 

80 864 

9.80 891 

9.80 919 

9.80 947 



80 975 
9.81 002 
9.81030 
9.81 058 
9.81 085 



9.81 118 
9.81 141 
9.81 168 
9.81 196 
9.81 224 



9.81 251 
Log. Cot. 



28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 

28 
28 
28 
28 
28 

27 
28 
28 
28 
28 

27 
28 
28 
28 
27 

28 
28 
27 
28 
28 

27 
28 
27 
28 
28 

27 
28 
27 
28 
27 
28 
27 
27 
28 
27 
28 
27 
27 
28 
27 
27 

cTd 



0.20 421 
0.20 393 
0.20 365 
0.20 337 
0.20 308 



0.20 140 
0.20 112 
0.20 084 
0.20 056 
0.20 028 



0-20 000 
0.19 972 
0.19 944 
0.19 916 
0.19 888 



0.19 860 
0.19 832 
0.19 804 
0.19 776 
0.19 748 9 



92 842 
92 834 
92 826 
92 818 
92 810 



.92 802 
.92 794 
• 92 786 
.92 778 
.92 771 



.92 763 
.92 755 
.92 747 
.92 739 
.92 731 



^92 723 
.92 715 
.92 707 
.92 699 
.92 691 



0.19 721 
0.19 693 
0.19 665 
0.19 637 
0^19 609 
0. 
0. 



19 581 
19 55^ 
19 525 
19 4^7 
19 470 



0.19 442 
0.19 414 
0.19 386 
0.19 358 
0-19 330 



0.19 303 
0.19 275 
0.19 247 
0.19 219 
0.19 191 



0.19 164 
0.19 136 
0.19 108 
0.19 080 
0-19 053 



0.19 
0.18 
0.18 
0-18 
0.18 



0.18 
0.18 
0.18 
0.18 
0.18 



025 
997 
970 
942 
914 
886 
859 
831 
803 
776 



0-18 748 
Log. Tan 



.92 683 
.92 675 
.92 667 
.92 659 
.92 651 



.92 643 
.92 635 
.92 6271 
.92 619 
• 92 611 



.92 603 
.92 595 
.92 587 
.92 579 
.92 570 



92 562 
92 554 
92 546 
92 538 
92 530 



92 522 
92 514 
92 506 
92 498 
92 489 



• 92 481 
.92 473 
.92 465 
.92 457 

• 92 449 



. 92 441 
.92 433 
.92 424 
.92 416 
. 92 408 



92 400 
92 392 
92 383 
92 375 
92 367 



9 92 359 
Log. Sin 



60 

59 
58 
57 
56^ 

55 
54 
53 
52 
11 
50 
49 
48 
47 
46 

45 
44 
43 
42 
41 
40 
39 
38 
37 
36^ 

35 
34 
33 
32 
11 
30 
29 
28 
27 
26 



20 

19 

18 

17 

16. 

15 

14 

13 

12 

JJL 

10 

9 

8 

7 

_6 

5 

4 

3 

2 

1 

O 



P.P. 





38 


38 


6 


2.8 


2.8 


7 


3.3 


3.2 


8 


3.8 


3.7 


9 


4.3 


4.2 


10 


4.7 


4.6 


20 


9.5 


9.3 


30 


14.2 


14.0 


40 


19.0 


18.6 


50 


23.7 


23.3 



3f„ 

2.7 

3.2 

36 

4.1 

4.6 

9.1 

13-7 

18.3 

22.9 



6 
7 
8 
9 

10 
20 
30 
40 
50 



30 30 



2.0 


2.0 


2.4 


2.3 


2.7 


2.6 


3.1 


3.0 


3.4 


3.3 


6.8 


6.6 


10.2 


10.0 


13.6 


13.3 


17.1 


16.6 



19. 

1.9 
2.3 
2.6 
2.9 
3.2 
6.5 
9.7 
13. Q 
16.2 



6 

7 
8 
9 
10 
20 
30 
40 
50 



8 

O.i 
l.C 
1.1 
1.3 
1-4 
2.8 
4.2 
5.6 
7.1 



1 _ 
2.6 
4.0 
5.3 
6.6 



0.^ 
0.9 
1.0 
1.1 
1.2 
2.5 
3.7 
5.0 
6.2 



P.P. 



133^ 



637 



Bf 



TABLE VII. 



83** 



-LOGARITHMIC SINES. COSINES, TANGENTS, 
AND COTANGENTS. 



146® 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 



9.73 
9-73 
9.73 
9.73 
9.73 



9.73 
9.73 
9.73 
9.73 
9.73 



9.73 
9.73 
9.73 
9.73 
9.73 



30 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

35 
36 
37 
88 
39. 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Log. Sin, 



9.73 
9.73 
9.73 
9.73 
9.73 



611 
630 
650 
669 
688 
708 
727 
746 
766 
785 
805 
824 
843 
862 
882 
901 
920 
940 
959 
978 



9.73 
9.74 
9.74 
9.74 
9.74 



9.74 
9.74 
9-74 
9.74 
9.74 



9.74 
9.74 
9.74 
9.74 
9.74 



9.74 
9-74 
9.74 
9.74 
9-74 
9.74 
9.74 
9.74 
9.74 
9.74 



9.74 
9.74 
9.74 
9.74 
9-74 



9.74 
9.74 
9.74 
9.74 
9.74 



9.74 
9.74 
9.74 
9.74 
9-74 



9-74 



Log. 



016 
036 
055 
074 
093 
112 
131 
151 
170 

189 
208 
227 
246 
265 
284 
303 
322 
341 
360 
379 
398 
417 
436 
455 
474 
493 
511 
535 
549 
568 
587 
606 
625 
643 
662 
681 
700 
718 
737 
756 
Cos. 



Log. Tan. c. d. Log. Cot. Log. Cos 



9.81 251 
9.81279 
9.81 307 
9.81334 
9.81 362 



9.81 390 
9.81 417 
9.81445 
9.81 473 
9.81 500 



81 528 
9.81 555 
9.81 583 
9.81 610 
9.81 638 



9.81 666 
9.81 693 
9.81 721 
81 748 
9.81 776 



9.81 803 
81 831 
9.81 858 
9-81 886 
9.81 913 



9.81 941 
9.81 968 

9.81 996 

9.82 023 
9-82 051 



9.82 078 

9.82 105 

9.82 133 

82 160 

82 188 




9.82 352 
9.82 380 
9.82 407 
9-82 434 
9.82 462 



9.82 489 
82 516 
9.82 544 
9.82 571 
9.82 598 



9.82 626 
82 653 
82 680 

9. 82 708 
82 735 



9-82 762 
9.82 789 
9-82 817 
9.82 844 
9. 82 871 



9-82 898 
Log. Cot, 



0.18 
0.18 
0.18 
0.18 
0.18 



0.18 
0.18 
0.18 
0.18 
0.18 



748 
720 
693 
665 
637 
610 
582 
555 
527 
499 



0.18 472 
0.18 444 
0.18 417 
0.18 389 
0.18 362 



0.18 334 
0.18 306 
0.18 279 
0.18 251 
0.18 224 



0.18 196 
0.18 169 
0.18 141 
0.18 114 
0.18 086 



0.18 059 
0.18 031 
0.18 004 
0.17 976 
0.17 949 



0.17 921 
0.17 894 
0.17 867 
0.17 839 
0.17 812 



0.17 784 
0.17 757 
0.17 729 
0.17 702 
0-17 675 



0.17 647 
0.17 620 
0.17 593 
17 565 
0.17 538 



0.17 510 
0.17 483 
0.17 456 
0.17 428 
0.17 401 



0.17 374 
0.17 347 
0.17 319 
0.17 292 
0.17 265 



0.17 237 
0.17 210 
0.17 183 
0.17 156 
0.17 128 



0.17 101 
Log. Tan, 




92 235 
92 227 
92 219 
92 210 
92 202 



92 194 
92 185 
92 177 
92 169 
92 160 



92 152 
92 144 
92 135 
92 127 
92 119 



92 110 
92 102 
92 094 
92 085 
92 077 



92 069 
92 060 
92 052 
92 043 
92 035 



92 027 
92 018 
92 010 
92 001 
91 993 



91 984 
91 976 
91 967 
91 959 
91 951 



91 942 
91 934 
91 925 
91 917 
91 908 



91 900 
91 891 
91 883 
91 874 
91 866 



9-91 857 
Log. Sin 



P.P. 





38 


37 


6 


2-8 


2.71 


7 


3 


2 


3 


2 


8 


3 


7 


3 


6 


9 


4 


2 


4 


1 


10 


4 


6 


4 


6 


20 


9 


3 


9 


1 


30 


14 





13 


7 


40 


18 


6 


18 


3 


50 


23 


3 


22 


9 



37 

2.7 

31 

3.6 

4.0 

4.5 

9.0 

13.5 

18-0 

22.5 





19 


19 


6 1.9 


1.91 


7 2.3 


2 


2 


8 


2.6 


2 


5 


9 


2.9 


2 


8 


10 


3.2 


3 


1 


20 


6-5 


6 


3 


30 


9.7 


9 


5 


40 


13.0 


12 


g 


50 


16.2 


15 


8 



18- 
1-8 
2-1 
2.4 
2.8 
3.1 
6.1 
9.2 
12-3 
15.4 





8 


8 


6 


0.8 


0.8 


7 


1.0 


0.9 


8 


1.1 




9 


1.3 


1-2 


10 


1.4 


1.3 


20 


2.8 


2.6 


30 


4.2 


4.0 


40 


5.6 


5.3 


50 


7.1 


6.6 



P. p. 



133^ 



038 



66^ 



34** 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



145* 



7 

8 

J_ 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 



Log, Sin 



9.74 756 
9-74 775 
9.74 793 
9.74 812 
9-74 831 



9.74 849 
9.74868 
9.74 887 
9.74 905 
9.74 924 



9.74 943 
9.74 961 
9.74 980 

9.74 998 

9.75 017 



20 

21 
22 
23 
124 
25 
26 
27 
28 
29, 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
i46 
'47 
48 
49 

50 

S5I 
'52 
,53 
54 
'55 
56 
57 
58 
59 
60 



9.75 036 
9.75 054 
9.75 073 
9.75 091 
9.75 110 



9.75 128 
9-75 147 
9.75 165 
9.75 184 
9.75202 



9.75 221 
9.75 239 
9.75 257 
9.75 276 
9.75 294 



9.75 313 
9.75331 
9.75349 
9.75 368 
9.75 386 



9.75 404 
9.75 423 
9.75441 
9.75459 
9-75 478 



9.75 496 
9.75 514 
9.75 532 
9.75 551 
9.75 569 



9.75 587 
9.75 605 
9-75 623 
9.75 642 
9-75 660 



9-75 678 
9-75 696 
75 714 
9-75 732 
9-75 750 



9^75 769 
9.75 787 
9.75 805 
9.75 823 
9.75 841 



9-75 859 



Log. Cos, 



Log. Tan. c. d. Log. Cot. Log. Cos 



9-82 898 
9.82 926 
9.82 953 

9.82 980 

9.8 3 007 
9.83 035 
9.83 062 
9.83 089 

83 116 
9-83 143 



9-83 307 
9.83 334 
9.83 361 
9.83 388 
9-83 415 



9-83 442 
9.83 469 
9-83 496 
9-83 524 
9-83 551 



9.83 578 
9-83 605 
9.83 632 
9-83 659 
9-83 686 



9-83 171 
9-83 198 
9-83 225 
9-83 252 
9.83 279 



9.83 713 
9-83 740 
9.83 767 
9.83 794 
9.83 821 



9.83 848 
9.83 875 
9.83 902 
9-83 929 
9-83 957 



9-83 984 
9-84011 
9-84 038 
9-84 065 
9.84 091 



9-84 118 
9-84 145 
9-84 172 
9-84 199 
9-84 226 



9-84 253 
9-84 280 
9-84 307 
9-84 334 
9.84 361 



9-84 388 
9-84 415 
9 - 84 442 
9-84 469 
9-84 496 



9-84 522 
Log. Cot, 



27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 

27 
27 
27 
27 
27 
27 
27 
27 
27 
27 

27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 
27 

27 
27 
27 
27 
26 

27 
27 
27 
27 
27 

27 
27 
27 
27 
26 
27 
27 
27 
27 
27 
26 

cTd 



0-17 101 
0.17 074 
0.17 047 
0.17 019 
0-16 99 2 
0.16 965 
0.16 938 
0.16 910 
0.16 883 
0.16 856 



0.16 422 
0.16 395 
0.16 368 
0.16 340 
0.16 313 



0.16 286 
0.16 259 
0.16 232 
0.16 2059 
0.16 178 



0.16 829 
0.16 802 
0.16 774 
0.16 747 
0.16720 



0.16 693 
0.16 666 
0.16 639 
0.16 612 
0-16 584 



9-91 857 
9-91 849 
9-91 840 
9-91 832 
9-91 823 
9.91 814 
9-91 806 
9-91 797 
9.91 789 
9-91780 



91 772 
91 763 
91 755 
91 746 
91737 



0.16 557 
0.16 530 
0.16 503 
0.16 476 
0.16 449 



9. 



91 729 
91 720 
91 712 
91 703 
91 694 



9.91 686 
9.91 677 
9.91 668 
9.91 660 
9.91 651 



9.91 642 
9.91 634 
9.91 625 
9.91 616 
9-91 608 



0.16 151 
0.16 124 
0.16 097 
0.16 070 
. 16 043 



0.16 016 
0.15 989 
0.15 962 
0.15 935 
0-15 908 



0.15 
0.15 
0-15 
0-15 
0-15 



881 

854 
827 
800 
773 



0-15 
0-15 
0.15 
0.15 
0-15 



746 
719 
699 
665 
639 



0.15 612 
0-15 585 
0.15 558 
0.15 531 
0.1 5 504 
0-15 477 
I Dg. Tan. 



91 599 
91 590 
91 582 
91 573 
91 564 



.91 556 
• 91 547 
-91 538 
-91 529 
•91 521 



91 512 
91 50§ 
91 495 
91486 
91 477 



91 468 
91460 

• 91451 

• 91442 
91433 



.91424 
.91416 
91407 
91 398 
91 389 



9-91 380 
9-91 372 
9-91 363 
9-91354 
9-91 345 



9-91 336 
Log. Sin 



40 

39 
38 

37 
11 
35 
34 
33 
32 
31 
30 
29 
28 
27 
21 
25 
24 
23 
22 
IL 
20 
19 
18 
17 
16 
15 
14 
13 
12 

n 

10 



p. p. 





27 


27 


6 


2-7 


2.7 


7 


3.2 


3.1 


8 


3.6 


3.6 


9 


4.1 


4.0 


10 


4.6 


4.5 


20 


9.1 


9-0 


30 


13.7 


13-5 


40 


18. 3 


18-0 


50 


22.9 


22.5 



26 

2.6 
31 
3.5 
40 

4.4 
8.8 

13.2 
17.6 
22.1 





19 


18 


6 


1.9 


1.8 


7 


2 


2 


2.1 


8 


2 


5 


2.4 


9 


2 


3 


2.8 


10 


3 


1 


3.1 


20 


6 


3 


6.1 


30 


9 


5 


9.2 


40 


12 




12.3 


50 


15 


8 


15.4 



18 

1.8 
2.1 
2.4 
2.7 
3.0 
6.0 
9.0 
12-0 
15.0 



6 

7 

8 

9 

10 

20 

30 

40 

50 






9 





1 





1 


1 


2 


1 


1 


3 


1- 


1 


5 


1- 


3 





2. 


4 


5 


4- 


6 





5. 


7 


5 


7. 



P. p. 



!1S4** 



639 



5SI' 



TABLE VIL- 



35° 



-LOGARITHMIC SINES. COSINES. TANGiSNTS, 
AND COTANGENTS. 



144° 



Log. Sin- 



Log. Tan. c. d. Log. Cot. Log. Cos 



P. P. 



75 859 
75 877 
75 895 
75 913 
75 931 



75 949 
75 967 

75 985 

76 003 
76 021 



76 039 
76 057 
76 075 
76 092 
76 110 



76 128 
76 146 
76 164 
76 182 
76 200 



76 217 
76 235 
76 253 
76 271 
76 289 



76 306 
76 324 
76 342 
76 360 
76 377 



76 395 
76 413 
76 431 
76 448 
76 466 



76 484 
76 501 
76 519 

75 536 

76 554 



76 572 
76 589 
76 607 
76 624 
76 642 



76 660 
76 677 
76 695 
76 712 
78 730 



76 747 
76 765 
76 782 
76 800 
76 817 



76 835 
76 852 
7,6 869 
76 887 
76 904 



9-76 922 



log, Cos, 



9-84 522 
9.84 549 
9.84 576 
9.84 603 
9.84 630 



84 657 
9.84 684 

84 711 
9.84 737 
9.84 764 



9.84 791 
9.84 818 
9.84 845 
9.84 871 
9.84 898 



9.84 925 
9.84 952 

9.84 979 

9.85 005 
9.85 032 



9.85 059 
9.85 086 
9.85 113 
9.85 139 
9.85 166 



9.85 193 
9.85 220 
9.85 246 
9.85 273 
9.85 300 



9.85 327 
9.85 353 
9.85 380 
9.85 407 
9.85 433 



9.85 460 
9.85 487 
9.85 513 
9.85 540 
9.85 567 



9.85 594 
9.85 620 
9.85 647 
9.85 673 
9.85 700 



9-85 727 
.85 753 
9.85 780 
9.85 807 
9-85 833 



9-85 860 
9.85 887 
9.85 913 
9.85 940 
9.85 966 



9.85 993 

9.86 020 
9.86 046 
9.86 073 
9-86 099 



9-86 126 
Log. Cot, 



0.15 477 
0-15 450 
0.15 423 
0.15 396 
0-15 370 



0.15 343 
0.15 316 
0.15 289 
0.15 262 
0.15 235 



0.15 208 
0.15 182 
0.15 155 
0.15 128 
0.15 101 



0.15 074 
0.15 048 
0.15 021 
0.14 994 
0.14 967 



0.14 940 
0.14 914 
0.14 887 
0.14 860 
0.14 833 



0.14 807 
0.14 780 
0.14 753 
0.14 726 
0.14 700 



0.14 673 
0.14 646 
0.14 620 
0.14 593 
0-14 566 



0.14 539 
0.14 513 
0-14 486 
0-14 459 
0-14 433 



0.14 406 
0.14379 
0.14353 
0.14 326 
0-14 299 



0-14 273 
0-14 246 
0-14219 
0.14193 
0-14 166 



0-14 140 
0-14113 
0-14086 
0-14 060 
0-14 033 



0-14 007 
0-13 980 
0-13 958 
0-13 927 
0-13 900 



0-13 874 
Log. Tan, 



91 336 
91 327 
91 318 
91310 
91 301 



91 292 
91 283 
91 274 
91 265 
91 256 



91 247 
91 239 
91 230 
91 221 
91 212 



91 203 
91 194 
91 185 
91 176 
91 167 



91 158 
91 149 
91 140 
91131 
91 122 



91 113 
91 104 
91095 
91 086 
91077 



91 068 
91059 
91050 
91041 
91032 



91 023 
91014 
91005 
90 996 
90 987 



90 978 
90 969 
90 960 
90 951 
90 942 



90 933 
90 923 
90 914 
90 905 
90 896 



90 887 
90 878 
90 869 
90 860 
90 850 



90 841 
90 832 
90 823 
90 814 
90 805 



1^5° 



9-90 796 
Log. Sin. 

"640 



27 


2-71 


3 


1 


3 


6 


4 





4 


5 


9 





13 


5 


18 





22 


5 



36. 

2-6 

3-1 

3-5 

4-0 

4-4 

8.8 

13.2 

17.6 

22-1 



6 

7 
8 
9 
10 
20 
30 
40 
50 



18 
1.8 
2.1 
2-4 
2.7 
3.0 
6.0 
9.0 
12.0 
15.0 



17_ 

1-7 
2.0 
2.3 
2-6 
2.9 
5.8 
8.7 
11.6 
14.6 



17 

1.7 

2 
5 
8 



P.P. 





9 


9 


8_ 


6 


0-9 


0.9 


0-8 


7 


1-1 


1.0 


1-0 


8 


1-2 


1-2 


l-I 


9 


1-4 


1-3 


1-3 


10 


1-6 


1-5 


1-4 


20 


3-1 


3-0 


2-8 


30 


4-7 


4.5 


4-2 


40 


6-3 


6.0 


5.6 


50 


7.9 


7.5 


71 



54° 



36" 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



143* 



5 
6 
7 
8 
_9^ 

10 

11 
12 
13 
Ik 
15 
16 
17 
18 
19 



9- 



Log» Sin, 



.76 922 

• 76 939 

• 76 956 

• 76 974 

• 76 991 



• 77 008 

• 77 026 

• 77 043 

• 77 060 

• 77 078 



77 095 
77 112 
77 130 
77 147 
77 164 



• 77 181 

• 77 198 

• 77 216 

• 77 233 

• 77 250 



20 
21 
22 
23 
2i 
25 
26 
27 
28 
29 
30 
31 
32 
33 
31 
35 
36 
37 
38 
39, 

40 

41 
42 
43 
44 
45 
46 
47 
48 
4i 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



• 77 267 

• 77 284 
•77 302 
•77 319 

• 77 336 



77 353 
77 370 
77 387 
77 404 
77 421 



77 439 
77 456 
77 473 
77 490 
77 507 



9-77 609 
9^77 626 
9.77 643 
9.77 660 
9.77 677 



9.77 693 
9.77 710 
9.77 727 
9 . 77 744 
9-77 761 



77 524 
77 541 
77 558 
77 575 
77 592 



9.77 778 
9.77 795 
9.77 812 
9.77 828 
9.77 845 



9.77 862 
9.77 879 
9.77 896 
9.77 913 
9.77 929 



9-77 946 



Log. Cos 



Log. Tan. c, d. Log. Cot. Log. Cos, 



9.86 126 
9.86 152 
9.86 179 
86 206 
9.88 232 



9-86 259 
9.86 285 
9.86 312 
9-86 338 
9.86 365 



9.86 391 
9-86 418 
9 •86 444 
9.86 471 
9^86 497 



9-86 524 

9^86 550 

86 577 

86 603 

9.86 630 



9.86 656 
9-86 683 
9-86 709 
9-86 736 
9-86 762 



9-86 788 
9.86 815 
9.86 841 
9-86 868 
9.86 894 



9-86 921 
9.86 947 
9-86 973 
9^87 000 
9^87 026 



9^87 053 
9-87 079 
9.87 105 
9.87 132 
9^87 158 



9-87 185 
9.87 211 
9-87 237 
9-87 264 
9.87 290 



9.87 316 
9.87 343 
9.87 369 
9.87 395 
9.87 422 



9.87 448 
9^87 474 
9^87 501 
9^87 527 
9.87 553 



9. 87 580 
9.87 60P 
9.87 632 
9.87 659 
9.87 685 



9-87 711 
Log. Cot, 



.13 874 

• 13 847 

• 13 821 

• 13 794 
•13 767 



• 13 741 
-13 714 
•13 688 
•13 661 
.13 635 



.13 608 
•13 582 
•13 555 
•13 529 
•13 502 



-13 476 
.13 449 
.13 423 
-13 396 
.13 370 



.90 657 
- 90 648 
• 90 63? 
-90 629 
.90 620 



9-90 611 
9-90 602 
9.90 592 
9-90 583 
9-90 574 



. 12 947 
.12 920 
.12 894 
.12 868 
-12 84l 



.12 815 
.12 78G 
.12 762 
.12 736 
-12 70S 



.12 68£ 
.12 657 
.12 63C 
• 12 60^. 
■12 578 
.12 551 
.12 52£ 
-12 499 
-12 472 
.12 446 



• 12 420 
-12 393 
.12 367 
-12 341 
.12 315 



0.12 288 



c.d. Log. Tan 



9-90 564 
90 555 
9-90 546 
9^90 536 
9-90 527 
9.90 518 
9-90 508 
9^90 499 
9-90 490 
9 - 90 480 



9 - 90 471 
9-90 461 
9^90 452 
9^90 443 
9-90 433 



9 . 90 424 
9.90 414 
9.90 405 
9.90 396 
9.90 386 



- Q 



9.90 377 
9.90 367 
9.90 358 
9 . 90 348 
9.90 339 



9.90 330 
9.90 320 
9.90 311 
9.90 301 
9.90 292 



90 282 

90 273 

9.90 263 

9-90 254 

90 244 



9-90 235 
Log. Sin. 



P. P. 



6 

7 
8 
9 
10 
20 
30 
40 
50 



27 

2.7 

3.1 

3.6 

4.0 

4.5 

9.0 

13.5 

18.0 

22.5 



36_ 

2.6 

3.1 

3.5 

4.0 

4.4 

8-8 

13.2 

17-6 

22.1 



26 

2^6 
3.0 

3.3 

3-9 

4-3 

86 

13. 

17.3 

21-6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



17 


17 


1-7 


1.7 


2.0 


2.0 


2.3 


2^2 


2.6 


2^5' 


2-9 


2.8 


5-8 


5.6 


8.7 


8.5 


11.6 


11.3 
14.1 


14.6 



16. 

1.6 
1.9 

2.2 
2.5 
2.7 
5.5 
8.2 
11.0 
13-7 





^ 


9 


6 


0.9 


0.9 


7 


1.1 


1 .0 


8 


1.2 


1.2 


9 


1-4 


1 .3 


10 


1-6 


1.5 


20 


3-1 


3.0 


30 


4.7 


4.5 


40 


6.3 


6.0 


50 


7.9 


7.5 



P.P. 



126° 



641 



63° 



TABLE VII.- 



37** 



-LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



143° 



' Log. Sin. d. Log. Tan. c.d. Log. Cot 



77 946 
77 963 
77 980 

77 996 

78 013 



78 030 
78 046 
78 063 
78 080 
78 097 



78 113 
78 130 
78 147 
78 163 
78 180 



78 196 
78 213 
78 230 
78 246 
78 263 



78 279 
78 296 
78 312 
78 329 
78 346 



78 362 
78 379 
78 395 
78 412 
78 428 



78 444 
78 461 
78 477 
78 494 
78 510 



78 527 
78 543 
78 559 
78 576 
78 592 



78 609 
78 625 
78 641 
78 658 
78 674 



78 690 
78 707 
78 723 
78 739 
78 755 



78 772 
78 788 
78 804 
78 821 
78 837 



78 853 
78 869 
78 885 
78 902 
78 918 
78 934 



Log. Cos, 



87 711 
87 737 
87 764 
87 790 
87816 



87 843 
87 869 
87 895 
87 92l 
87 948 



87 974 

88 000 
88 026 
88 053 
88 079 



88 105 
88 13l 
88 157 
88 184 
88 210 



88 236 
88 262 
88 288 
88 315 
88 341 



88 367 
88 393 
88 419 
88 445 
88 472 



88 498 
88 524 
88 550 
88 576 
88 602 



88 629 
88 655 
88 681 
88 707 
88 733 



88 759 
88 785 
88 8ll 
88 838 
88 864 




Log. Cot. 



26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

cTJ 



0.12 288 
0-12 262 
0.12 236 
0.12 209 
0.12 183 



0.12 
0.12 
0-12 
0-12 
0.12 



0.12 
0.11 
0.11 
0.11 
0.11 



157 
131 
104 
078 
052 

026 
999 
973 
947 
921 



0.11 895 
0.11868 
0.11 842 
0.11 816 
0.11 790 



, Cos. d. 



0.11 76S 9 
0.11 737 9 
0.11 711 
0.11 685 
0.11 659 



0-11 633 
0.11 606 
0.11 580 
0.11 554 
0.11 528 



0-11 502 
0.11476 
0.11 449 
0.11 428 
0.11 397 



0.11 371 
0.11 345 
0.11319 
0.11 293 
0.11 266 



0.11 
0.11 
0.11 
0.11 
0.11 



0.11 
0.11 
0.11 
0.11 
0.11 



240 
214 
188 
162 
136 
110 
084 
058 
032 
005 



0.10 979 
0.10 953 
0.10 927 
0.10 901 
0.10 875 



• 10 849 

.10 823 

0.10 797 

0.10 771 

10 745 



0-10 719 



Log. Tan. 



90 235 
90 225 
90 216 
90 206 
90 196 



90 187 
90 177 
90 168 
90 158 
90 149 



90 139 
90 130 
90 120 
90 110 
90 101 



90091 
90 082 
90 072 
90 062 
90 053 



90 043 
90 033 
90 024 
90 014 
90 004 



89 995 
89 985 
89 975 
89 966 
89 956 



89 946 
89 937 
89 927 
89 917 
89 908 



89 898 
89 888 
89 878 
89 869 
89 859 



89 849 
89 839 
89 830 
89 820 
89 810 



89 800 
89 791 
89 781 
89 771 
89 761 



89 751 
89 742 
89 732 
89 722 
89 712 



89 702 
89 692 
89 683 
89 673 
89 663 



89 653 



-og. Sin. 



P.P. 



6 

7 
8 
9 
10 
20 
30 
40 
50 



26 



2.61 


3 


1 


3 


5 


4 





4 


4 


8 


8 


13 


2 


17 


6 


22 


1 



26 

2.6 

3.0 

3-4 

3.9 

4.3 

8.6 

13.0 

17.3 

21.6 





17 


16 


6 


1.7 


1.6 


7 


2.0 


1.9 


8 


2.2 


2.2 


9 


2.5 


2.5 


10 


2-8 


2.7 


20 


5.6 


5.5 


30 


8.5 


8.2 


40 


11.3 


11. C 


50 


14.1 


13.7 



16 

1.6 
1.8 
2.1 
2.4 
2.6 
5.3 
8.0 
10.6 
13.3 



10 9 



61 





0.9 


7|l 


1 


1.1 


81 


3 


1.2 


91 


5 


1.4 


10 1 


6 


1.6 


20 3 


33.1 


30 5 


04-7 


40 6 


66.3 


50 8 


3 


7.9 



P. p. 



127** 



%i2 



62° 



TABLE VII. 



-LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



141® 



Log. Sin. 



78 934 
78 950 
78 966 
78 982 
78 999 



79 015 
79 031 
79 047 
79 063 
79 079 



79 095 
79 111 
79 127 
79 143 
79 159 
79 175 
79 191 
79 207 
79 223 
79 239 



79 255 
79 271 
79 287 
79 303 
79 319 



79 335 
79 351 
79 367 
79 383 
79 399 



79 415 
79 431 
79 446 
79 462 
79 478 



79 494 
79 510 
79 526 
79 541 
79 557 



79 573 
79 589 
79 605 
79 620 
79 636 



79 652 
79 668 
79 683 
79 699 
79 715 



79 730 
79 746 
79 762 
79 777 
79 793 



79 809 
79 824 
79 840 
79 856 
79 871 



9-79 887 
Log. Cos. 



d. Log. Tan. c.d. Log. Cot. Log. Cos. 



89 281 
89 307 
89 333 
89 359 
89 385 



89 411 
89 437 
89 463 
89 489 
89 515 



89 541 
89 567 
89 593 
89 619 
89 645 



89 671 
89 697 
89 723 
89 749 
89 775 



89 801 
89 827 
89 853 
89 879 
89 905 



89 931 
89 957 

89 982 

90 008 
90 034 



90 060 
90 086 
90 112 
90 138 
90164 



90 190 
90 216 
90 242 
90 268 
90 294 



90 319 
90 345 
90 371 
90 397 
90 423 



90 449 
90 475 
90 501 
90 526 
90 552 



90 578 
90 604 
90 630 
90 656 
90 682 



90 707 
90 733 
90 759 
90 785 
90 811 



9.90 837 
Log. Cot. 



26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
25 
26 
26 
26 
26 
26 
26 
25 
26 
26 
26 
26 
26 
25 
26 
26 
26 
25 
26 
26 
26 
25 
26 
26 
26 
25 
26 
26 
25 
26 
26 
26 
25 
26 

cl. 



10 719 
10 693 
10 667 
10 641 
10 615 



10 539 
10 563 
10 537 
10 511 
10 485 



10 459 
10 433 
10 407 
10 381 
10 355 



10 329 
10 303 
10 277 
10 251 
10 225 



10 199 
10 173 
10 147 
10 121 
10 095 



10 069 
10 043 
10 017 
09 991 
09 965 



09 939 
09 913 
09 887 
09 861 
9 83^ 
09 810 
09 784 
09 758 
09 732 
09 706 



09 680 
09 654 
09 628 
09 602 
09 577 



09 551 
09 525 
09 499 
09 473 
09 447 



09 421 
09 395 
09 370 
09 344 
09 318 



09 292 
09 266 
09 240 
09 214 
09 189 



0-09 163 
Log. Tan. 



89 653 
89 643 
89 633 
89 623 
89 613 



89 604 
89 594 
89 584 
89 574 
89 564 



89 554 
89 544 
89 534 
89 524 
89 514 



89 504 
89 494 
89 484 
89 474 
89 464 



89 454 
89 444 
89 434 
89 424 
89 414 



89 404 
89 394 
89 384 
89 374 
89 364 



89 354 
89 344 
89 334 
89 324 
89 314 



89 304 
89 294 
89 284 
89 274 
89 264 



89 253 
89 243 
89 233 
89 223 
89 213 



89 203 
89 193 
89 182 
89 172 
89 162 



89 152 
89 142 
89 132 
89 121 
89 111 



89 101 
89 091 
89 081 
89 070 
89 060 



89 050 



Log. Sin. d, 
643 



60 

59 
58 
57 
56 
55 
54 
53 
52 
51 

50 

49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 
38 
37 
_36 
35 
34 
33 
32 
31. 
30 
29 
28 
27 
26, 
25 
24 
23 
22 
21 



P.P. 



36 

6 2.6 

7 3.0 

8 3.i 

9 39 
10 4.3 
20 8-6 
30 13.0 
40117.3 
50121.6 



35 

2.5 

3.0 

3.4 

3.8 

4.2 

8-5 

12.7 

17.0 

21.2 





16 


16 


15 


6 


1.6 


1-6 


1.5 


7 


1.9 


1 


8 


1 


8 


8 


2.2 


2 


1 


2 





9 


2.5 


2 


4 


2 


3 


10 


2-7 


2 


6 


2 


6 


20 


5-5 


5 


3 


5 


1 


30 


8-2 


8 





7 


7 


40 


11.0 


10 


6 


10 


3 


50 


13-7 


13 


3 


12 


9 



10 10 9 



6 


1 





1 








9 


7 




2 




1 


1 


1 


8 




4 




3 


1 


2 


9 




6 




5 


1 


4 


10 




7 




6 


1 


6 


20 


3 


5 


3 


3 


3 


1 


30 


5 


2 


5 





4 


7 


40 


7 





6 


6 


6 


3 


50 


8 


7 


8 


3 


7 


9 



P. p. 



51' 



39' 



TABLE VIL— LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



140** 



' Log. Sin. d. Log, Tan. c.d. Log. Cot. Log. Cos, d 



O 

1 
2 
3 

5 

6 

7 

8 

_9_ 
10 
11 
12 
13 
li 
15 
16 
17 
18 
19 



79 887 
79 903 
79 918 
79 934 
79 949 



79 965 
79 980 

79 996 

80 011 
80 027 



80 042 
80 058 
80 073 
80 089 
80 104 



80 120 
80 135 
80 151 
80 166 
80 182 



20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 
36 
37 
38 
39 



80 197 
80 213 
80 228 
80 243 
80 259 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
5i 
60 



• 80 274 

• 80 289 
80 305 

-80 320 
80 335 



1.80 351 

• 80 366 

• 80 381 
1-80 397 
1.80 412 



.80 427 
. 80 443 

80 458 
. 80 473 

80 488 



.80 504 
.80 519 
.80 534 

• 80 549 

• 80 564 



1.80 580 
1.80 595 
1.80 610 
1. 80 625 
1 . 80 640 



80 655 
80 671 
80 686 
80 701 
80 716 



■ 80 731 

• 80 746 
•80 761 

80 776 

• 80 791 



9 80 806 



Log. Cos. d 



90 837 
90 863 
90 888 
90 914 
90 940 



90 966 

90 992 

91 017 
91043 
91 069 



91095 
91 121 
91 146 
91 172 
91 198 



91 224 
91 250 
91 275 
91 301 
91 327 



91 353 
91378 
91 404 
91430 
91456 



91481 
91 507 
91 533 
91 559 
91 584 



91 610 
91 636 
91 662 
91 687 
91 713 



91 739 
91 765 
91 790 
91 816 
91 842 



91 867 
91 893 
91 919 
91945 
91 970 



91 996 

92 022 
92 047 
92 073 
92 099 



92 124 
92 150 
92 176 
92 201 
92 227 



92 253 
92 278 
92 304 
92 330 
92 355 



9^92 381 
Log. Cot, 



09 163 
09 137 
09 111 
09 085 
09 060 



09 034 
09 008 
08 982 
08 956 
08 930 



08 905 
08 879 
08 853 
08 827 
08 802 



08 776 
08 750 
08 724 
08 698 
08 673 



08 647 
08 621 
08 595 
08 570 
08 544 



08 518 
08 492 
08 467 
08 441 
08 415 





07 875 
07 849 
07 824 
07 798 
07 772 



07 747 
07 721 
07 695 
07 670 
07 644 



07 61 G 



Log. Tan. 



89 050 
89 040 
89 030 
89 019 
89 009 



88 999 
88 989 
88 978 
88 968 
88 958 



88 947 
88 937 
88 927 
88 917 
88 906 



88 896 
88 886 
88 875 
88 865 
88 855 



88 844 
88 834 
88 823 
88 813 
88 803 



88 792 
88 782 
88 772 
88 761 
88 751 



88 740 
88 730 
88 720 
88 709 
88 699 



88 688 
88 678 
88 667 
88 657 
88 646 



88 636 
88 625 
88 615 
88 604 
88 594 



88 583 
88 573 
88 562 
88 552 
88 54l 



88 531 
88 520 
88 510 
88 499 
88 489 



88 478 
88 467 
88 457 
88 446 
88 436 
9 •88 425 
Log. Sin. 



P. p. 



36 



2 


6 


2. 


3 





3. 


3 


4 


3- 


3 


9 


3. 


4 


3 


4. 


8 


6 


8. 


13 





12. 


17 


3 


17. 


21 


6 


21. 



25 

5 

4 
8 
2 
5 
7 






16 


15 


15 


6 


1.6 


1.5 


1 5 


7 


1 


8 


1-8 


1 


7 


8 


2 


1 


2.0 


2 





9 


2 


4 


2-3 


2 


2 


10 


2 




2.6 


2 


5 


20 


5 


3 


5.1 


5 





30 


8 





7.7 


7 


5 


40 


10 


6 


10.3 


10 





50 


13 


3 


12.9 


12 


5 





11 


10 


10 


6 


1.1 


1.0 


1.0 


7 




3 




2 




1 


8 




4 




4 




3 


9 




6 




6 


1 


5 


10 




8 




7 




6 


20 


3 


6 


3 


5 


3 


3 


30 


5 


5 


5 


2 


5 





40 


7 


3 


7 


06 6 


50 


9 


1 


8 


7 


8 


3 



P.P. 



laG** 



644 



50° 



TABLE VII.— LOGARITHMIC SINES, COSINES, TANGENTS. 
AND COTANGENTS. 



139° 



Log. Sin 



9 



80 806 
80 822 
80 837 
80 852 
80 867 



80 882 
80 897 
80 912 
80 927 
80 942 



80 957 
80 972 

80 987 

81 001 
81 016 



81031 
81 046 
81061 
81076 
81 091 



81 106 
81 121 
81 136 
81 150 
81 165 



81 180 
81 195 
81210 
81 225 
81 239 



81 254 
81 269 
81284 
81 299 
81 313 



81 328 
81 343 
81358 
81 372 
81 387 



81 402 
81416 
81431 
81446 
81 460 



81475 
81490 
81 504 
81 519 
81 534 



81 548 
81 563 
81 578 
81 592 
81 607 



81 621 
81 636 
81 650 
81 665 
81 680 



9-81 694 



d. Log, Tan. c. d. Log. Cot 



93 022 
93 047 
93 073 
93 098 
93 124 



92 381 
92 407 
92 432 
92 458 
92 484 



92 509 
92 535 
92 561 
92 586 
92 612 



92 638 
92 663 
92 689 
92 714 
92 740 



92 766 
92 791 
92 817 
92 842 
92 838 



OS 



92 894 
92 919 
92 945 
92 971 
92 996 



93 150 
93 175 
93 201 
93 226 
93 252 



93 278 
93 303 
93 329 
93 354 
93 380 



93 405 
93 431 
93 456 
93 482 
93 508 



93 533 
93 559 
93 584 
93 610 
93 635 



93 661 
93 686 
93 712 
93 737 
93 763 



93 788 
93 814 
93*840 
93 865 
93 891 



93 916 



07 618 
07 593 
07 567 
07 541 
07 516 



Log. Cos. d. 



07 490 
07 465 
07 439 
07 413 
07 388 



07 362 
07 336 
07 311 
07 285 
07 259 



07 234 
07 208 
07 183 
07 157 
07 131 



07 106 
07 080 
07 055 
07 029 
07 003 



06 978 
06 952 
06 927 
06 901 
06 875 



06 850 
06 824 
06 799 
06 773 
06 748 



06 722 
06 696 
06 671 
06 645 
06 620 



06 594 
06 569 
06 543 
06 518 
06 492 



06 466 
06 441 
06 415 
06 390 
06 364 



06 339 
06 313 
06 288 
06 262 
06 237 



06 211 
06 186 
06 16C 
06 134 
06 109 



0-06 083 



88 425 
88 415 
88 404 
88 393 
88 383 



88 372 
88 361 
88 351 
88 340 
88 329 



88 319 
88 308 
88 297 
88 287 
88 276 



88 265 
88 255 
88 244 
88 233 
88 223 



88 212 
88 201 
88 190 
88 180 
88 169 



88 158 
88 147 
88 137 
88 126 
88 115 



88 104 
88 094 
88 083 
88 072 
88 06l 



88 050 
88 039 
88 029 
88 018 
88 007 



87 996 
87 985 
87 974 
87 963 
87 953 



87 942 
87 931 
87 920 
87 909 
87 898 



87 887 
87 876 
87 865 
87 854 
87 844 



87 833 
87 822 
87 811 
87 800 
87 789 



9-87 778 






P.P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



36 



2 


6 


2 


3 





3 


3 


4 


3 


3 


9 


3 


4 


3 


4 


8 


6 


8 


13 





12. 


17 


3 


17. 


21 


6 


21. 



25 

5 

4 
8 
2 
5 
7 

2 



6 

7 

8 

9 

10 

20 

30 

40 

50 



15 


15 


1^ 


1.5 


1.5 


1. 


1 


8 


1 


7 


1. 


2 





2 





1. 


2 


3 


2 


2 


2. 


2 


6 


2 


5 


2. 


5 


1 


5 





4. 


7 


7 


7 


5 


7. 


10 


3 


10 





9. 


12 


9 


12 


5 


12. 



11 



lO 



6 


1 


1 


l.S 


7 




3 


1.2 


8 




4 


1.4 


9 




6 


1.6 


10 




8 


1.7 


20 


3 


6 


3.5 


30 


5 


5 


5.2 


40 


7 


3 


7.0 


50 


9 


1 


8.7 



Log, Cos, 



d. Log. Cot. c. d.jLog. Tan 



Log. Sin. 



P. P. 



645 



49^ 



41° 



TABLE VIL—LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



138** 



10 

11 
12 
13 
li 
15 
16 
17 
18 
19 



Log. Sin 



9.81 694 
9.81 709 
9.81 723 
81 738 
9.81 752 



81 767 
9.81781 
9-81 796 
9.81 810 
9.81 824 



9.81 839 
9.81 853 
9. 81 868 
9. 81 882 
9. 81 897 



20 

2J 
22 
23 
24 
25 
26 
27 
28 
2i 
30 
31 
32 
33 
34 



9.81 911 
9.81 925 
9.81 940 
9.81 954 
9-81 969 



81 983 

9.81 997 

9.82 012 

82 026 
9.82 040 

9.82 055 
9.82 069 
9.82 083 
82 098 
9.82 112 



9-82 126 
9.82 140 
82 155 
9.82 169 
9.82 183 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49^ 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59. 
60 



9.82 197 
9.82 212 
9.82 226 
9 . 82 240 
9.82 254 



82 269 
82 283 
82 297 
82 311 
82 325 



82 339 
82 354 
82 368 
82 382 
82 396 



.82 410 
.82 424 
■ 82 438 
82 452 
• 82 467 



•82 481 

• 82 495 
82 509 

• 82 523 

• 82 537 



9-82 551 



Log, Cos, 



14 
14 
li 
14 

14 
14 

14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 

14 
14 
14 
ll 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 



Log. Tan. c. d 



9.93 916 
9.93 942 
9.93 967 

9.93 993 

9.94 018 
9.94 044 
9.94 069 
9.94 095 
9.94 120 

94 146 



9.94 171 
9.94 197 
9.94 222 
9.94 248 
9.94 273 



9.94 299 
9.94 324 
9.94 350 
9.94 375 
9 . 94 400 



9.94 426 
9.94 451 
9.94 477 
9.94 502 
9.94 528 



9.94 553 
9.94 579 
9 . 94 604 
9.94 630 
9.94 655 



9.94 681 
9.94 706 
9-94 732 
9.94 757 
9.94 782 



9.94 808 
9.94 833 
9.94 859 
9.94 884 
9.94 910 



9.94 935 
9.94 961 

9.94 986 

9.95 011 
9.95 037 



9.95 062 
9.95 088 
9.95 113 
9.95 139 
9.95 164 



9.95 189 
9.95 215 
9.95 240 
9.95 266 
9. 95 291 



9.95 316 
9.95 342 
9.95 367 
9. 95 393 
9.95 418 



9^95 443 



Log. Cot. c. d 



25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 

25 



Log. Cot. Log. Cos 



0.06 083 

06 058 

06 032 

0.06 007 

0.05 981 



0.05 956 
0.05 930 
0.05 905 
0-05 879 
0-05 854 



0-05 828 
0.05 803 
0.05 777 
0.05 752 
0.0572 6 



0.05 701 
0-05 675 
0-05 650 
0.05 625 
0.05 599 



0.05 574 
005 548 
0-05 523 
0.05 497 
0.05 472 



0.05 446 
0.05 421 
0.05 395 
005 370 
0.05 344 



0-05 319 
0.05 293 
005 268 
0.05 243 
0.05 217 



9.87 778 
9-87 767 
87 756 
9.87 745 
9.87 734 



9.87 723 
9.87 712 
9. 87 701 
9.87 690 
9-87 679 



9.87 668 
9.87 657 
9.87 645 
9-87 634 
9-87 623 



9-87 612 
9-87 601 
9-87 590 
9.87 579 
9 87 568 



9.87 557 
9.87 546 
9.87 535 
9-87 523 
9-87 512 



9-87 501 
9- 87 490 
9.87 479 
9. 87 468 
9-87 457 



0.05 192 
0-05 166 
0.05 141 
0-05 115 
0.05 090 



0.05 064 
0.05 039 
0.05 014 
0-04 988 
0- 04 963 



0.04 810 
0.04 785 
0.04 759 
0.04 734 
. 04 708 



0.04 937 
0.04 912 
0.04 886 
0.04 861 
0.04 836 



9.87 445 
9-87 434 
9-87 423 
9.87 412 
9.87 401 



9.87 389 
9-87 378 
9.87 367 
9-87 356 
9-87 345 



9-87 333 
9-87 322 
9-87 311 
9-87 300 
9 -87 288 



9-87 221 
9.87 209 
9.87 198 
9.87 187 
9-87 175 



0.04 683 
0-04 658 
0^ 04 632 
. 04 607 
04 581 



0-04 556 



Log. Tan. 



9.87 277 
9-87 266 
9-87 254 
9-87 243 
9-87 232 



9-87 164 
9. 87 153 
9-87 141 
9.87 130 
87 118 



9-87 107 



Log. Sin. d. 



11 
11 
11 
11 

11 
11 
11 
11 
11 
11 
11 
ll 
11 
11 
11 
11 
11 
11 
11 
11 
11 
11 
11 
11 
11 
11 
11 
11 
11 
ll 
11 
ll 
11 
11 
ll 
ll 
ll 

11 

11 

ll 

11 

11 

11 

ll 

ll 

11 

ll 

11 

ll 

11 

11 

11 

11 

11 

ll 

11 

11 

11 

11 

11 



60 

59 
58 

57 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46^ 

45 
44 
43 
42 
41 

40 

39 
38 

37 

li 

35 
34 
33 
32 
IL 
30 
29 
28 
27 
21 
25 
24 
23 
22 

30 

19 
18 
17 
16 



P.P. 





35 


25 


6 


2.5 


2.5 


7 


3 





2 


9 


8 


3 


4 


3 


3 


9 


3 


8 


3 


7 


10 


4 


2 


4 


1 


20 


8 


5 


8 


3 


30 


12 


7 


12 


5 


40 


17 





16 




50 


21 


2 


20 


8 





11 


14 


6 


1-4 


1.4 


7 


1 


7 


1 6 


8 


1 


9 


]-8 


9 


2 


2 


2-1 


10 


2 


4 


2-.S 


20 


4 


8 


4-6 


30 


7 


2 


7-0 


40 


9 


6 


9.3 


50 


12 


1 


11.6 





11 


11 


6 


1-1 


1-1 


7 




3 


1-3 


8 




5 


1 -4 


9 




7 


1 - 6 


10 




9 


1 -8 


20 


3 


8 


3.6 


30 


5 


7 


5.5 


40 


7 


6 


7.3 


50 


9 


6 


9.1 



P.P. 



ISl** 



646 



48^ 



43° 



TABLE VII.— LOGARITHMIC SINES. COSINES, TANGENTS, 
AND COTANGENTS. 



5 
6 
7 
8 

10 

11 
12 
13 
11 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29. 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Log. Sin. 



82 551 
82 565 
82 579 
82 593 
82 607 



82 621 
82 635 
82 649 
82 663 
82 677 



82 691 
82 705 
82 719| 
82 7331 
82 746j 



82 760 
82 774l 
82 788 
82 802 
82 816 



82 830 
82 844 
82 858i 
82 871 
82 885 



82 899 
82 913 
82 927 
82 940 
82 954 



82 968 
82 982 

82 996 

83 009 
83 023 



83 037 
83 051 
83 064 
83 078 
83 092 



83 106 
83 119 
83 133 
83 147 
83 160 



83 174 
83 188 
83 201 
83 215 
83 229 



83 242 
83 256 
83 269 
83 283 
83297 



83 310 
83 324 
83 337 
83 351 
83 365 



983 378 



Log, Cos. 



Log. Tafi.iC d, 



95 443 
95 469 
95 494 
95 520 
95 545 



95 571 
95 596 
95 621 
95 647 
95 672 



95 697 
95 723 
95 748 
95 774 
95 799 



95 824 
95 850 
95 875 
95 901 
95 926 



95 951 

95 977 

96 002 
96 027 
96 053 



96 078 
96 104 
96 129 
96 154 
96 180 



96 205 
96 230 
96 256 
96 281 
96 306 



96 332 
96 357 
96 383 
96 408 
96 433 



96 459 
96 484 
96 509 
96 535 
96 560 



96 585 
96 611 
96 636 
96 661 
96 687 



96 712 
96 737 
96 763 
96 788 
96 813 



96 839 
96 864 
96 889 
96 915 
9 6 940 
96 965 
Log. Cot, 



Log. Cot. Log. Cos 



04 556 
04 531 
04 505 
04 48Q 
04 454 



04 429 
04 404 
04 378 
04 353 
04 327 



04 302 
04 277 
04 251 
04 226 
04 200 



04 175 
04 150 
04 124 
04 099 
04 074 



04 048 
04 023 
03 997 
03 972 
03 947 



921 
896 
871 
845 
820 
795 
769 
744 
718 
693 



03 668 
03 642 
03 617 
03 592 
03 566 



03 541 
03 516 
03 490 
03 465 
03 440 



03 414 
03 389 
03 364 
03 338 
03 313 



03 287 
03 262 
03 237 
03 211 
03 186 



03 161 
03 135 
03 110 
03 085 
03 059 



0.03034 



Log, Tan, 



^^7 050 
S; 039 
8f 027 
87 016 
87 004 



87 107 
87 096 
87 084 
87 073 
87 062 



86 993 
86 982 
86 970 
86 959 
86 947 



86 936 
86 924 
86 913 
86 901 
86 890 



86 878 
86 867 
86 855 
86 844 
86 832 



86 821 
86 809 
86 798 
86 786 
86 774 



86 763 
86 751 
86 740 
86 728 
86716 



86 705 
86 693 
86 682 
86 670 
86 658 



86 647 
86 635 

85 623 

86 612 
86 600 



86 588 
86 577 
86 565 
86 553 
86 542 



86 530 
86 518 
86 507 
86 495 
86 488 



86 471 
86 460 
86 448 
86 436 
86 424 



9-86 412 
Log. Sin, 



P. P. 



25. 

2-5 



10 4 
20 8 
30il2 
40il7 
50121 



25 

2-5 

2 

3 

3 

4 



14 


13 


1.4 


1.3 


1 


6 


1 


6 


1 


8 


1 


8 


2 


1 


2 





2 


3 


2 


2 


4 


6 


4 


5 


7 





6 


7 


9 


3 


9 




11 


6 


11 


2 



12 

1.2 

1 
1 
1 
2 
4 



11 11 

1.1:1.1 

3 1.3 
5,1.4 

7 1.6 
9 1-8 

8 3-6 
7 5-5 
617.3 
619.1 



P. P. 



132^ 



647 



47^ 



43° 



TABLE VII.—LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



136'' 



10 

11 
12 
13 

15 
16 
17 
18 
19 



9.83 378 
9.83 392 
9.83 405 
9.83 419 
9. 83 432 



9 . 83 446 
9.83 459 
9.83 473 
9.83 486 
9.83 500 



9.83 513 
9.83 527 
9.83 540 
9. 83 554 
9.83 567 



30 

21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 



Log. Sin. 



9.83 580 
9.83 594 
9.83 607 
9.83 621 
9. 83 634 



9.83 647 

9.83 661 

83 674 

83 688 

9.83 701 



9.83 714 
9.83 728 
9.83 741 
9.83 754 
9.83 768 



83 781 
9.83 794 
9. 83 808 
9.83 821 
9.83 834 



9.83 847 
9.83 861 
9.83 874 
9. 83 887 
9-83 900 



40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

5i 

60 



9-83 914 
9.83 927 
9.83 940 
9.83 953 
9.83 967 



9.83 980 

9.83 993 

9.84 006 
9.84 019 
9.84 033 



9.84 046 
9.84 059 
9.84 072 
9.84 085 
9.84 098 




Log. Cos 



d. 



Log. Tan. c. d. Log. Cot. Log. Cos 



96 965 
9.96991 
9.97 016 
9.97 041 
9.97 067 



9.97 092 
9.97 117 
9.97 143 
9.97 168 
9.97 193 



9.97 219 
9.97 244 
9.97 269 
9.97 295 
9.97 320 




9.97 598 
9.97 624 
9.97 649 
9.97 674 
9.97 699 



97 725 

97 750 

9.97 775 

9.97 801 

9.97 826 



9. 97 851 
9.97 877 
9-97 902 
9-97 927 
97 952 



9.97 978 

9.98 003 
9.98 028 
9.98 054 
9.98 079 



98 104 
9.98 129 
9.98 155 
9.98 180 
9.98 205 



9.98 231 
9.98 256 
9-98 281 
9.98 306 
9.98 332 



9.98 357 
9.98 382 
9.98 408 
9.98 433 
9.98^58 
9.98 483 
Log. Cot. 



0.03 034 
0.03 009 
0.02 984 
0.02 958 
0.02 933 



0.02 908 
0.02 882 
0.02 857 
0.02 832 
0.02 806 



0.02 781 
0.02 756 
0.02 730 
0.02 705 
0.02 680 



0.02 654 
0.02 629 
0.02 604 
0.02 578 
0.02 553 



0.02 528 
0.02 502 
0.02 477 
0.02 452 
0.02 427 



0.02 401 
0.02 376 
0.02 351 
0.02 325 
0.02 300 



0.02 275 
0.02 249 
0.02 224 
0.02 199 
0.02 174 



0.02 148 
0.02 123 
0.02 098 
0-02 072 
0.02 047 



0.02 022 
0.01 996 
0.01 971 
0.01 946 
0.01 921 



0.01 895 
0.01 87C 
0.01 845 
0.01 819 
O-Ol 794 



0.01 769 
0.01 744 
0.01 718 
0-01 693 
0.01 668 




Log. Tan. 



86 412 
86 401 
86 389 
86 377 
86 365 



86 354 
86 342 
86 330 
86 318 
86 306 



86 294 
86 282 
86 271 
86 259 
86 247 



86 235 
86 223 
86 211 
86 199 
86 187 



86 176 
86 164 
86 152 
86 140 
86 128 



86 116 
86 104 
86 092 
86 080 
86 068 



86 056 
86 044 
86 032 
86 020 
86 008 



85 996 
85 984 
85 972 
85 960 
85 948 



85 936 
85 924 
85 912 
85 900 
85 887 



85 875 
85 863 
85 851 
85 839 
85 827 



85 815 
85 803 
85 791 
85 778 
85 766 



85 754 
85 742 
85 730 
85 718 
SJLZPS 
85 69"§ 



133° 



Log. Sin, 
648 



P.P. 



25 



2 


5 


2. 


3 





2. 


3 


4 


3. 


3 


8 


3. 


4 


2 


4. 


8 


5 


8. 


12 


7 


12. 


17 





16. 


21 


2 


20. 



35 

5 
9 
3 
7 
I 
3 
5 
6 
8 





13 


13 


6 


1.3 


1.3 


7 


1 


6 


1 


5 


8 


1 


8 


1 


7 


9 


2 





1 


9 


10 


2 


2 


2 


1 


20 


4 


5 


4 


3 


30 


6 


7 


6 


5 


40 


9 





8 


3 


50 


11 


2 


10 


8 





13 


13 


IT 


6 


1.2 


1.2 


1.1 


7 


1 


4 


1 


4 




3 


8 


1 


6 


1 


6 




5 


9 


1 


9 


1 


8 




7 


10 


2 


1 


2 







9 


20 


4 


1 


4 





3 


s 


30 


6 


2 


6 


05 


7 


40 


8 


3 


8 


7 


Q 


50 


10 


4 


10 





9 


6 



P. p. 



46° 



44° 



TABLE VII. —LOGARITHMIC SINES, COSINES, TANGENTS, 
AND COTANGENTS. 



135° 



Log, Sin, 



84 177 
84 190 
84 203 
84 216 
84 229 



84 242 
84 255 
84 268 
84 281 
84 294 



84 307 
84 320 
84 333 
84 346 
84 359 



84 372 
84 385 
84 398 
84 411 

84 424 



84 437 
84 450 
84 463 
84 476 
84 489 



84 502 
84 514 
84 527 
84 540 
84 553 



84 566 
84 579 
84 592 
84 604 
84 617 



84 630 
84 643 
84 656 
84 669 
84 681 



84 694 
84 707 
84 720 
84 732 
84 745 



84 758 

84 771 
84 783 
84 796 
84 809 



84 822 
84 834 
84 847 
84 860 
84 872 



84 885 
84 898 
84 910 
84 923 
84 936 



9-84 948 



Log. Cos. 



Log. Tan. c.d. Log. Cot. Log. Cos. 



99 



99 



483 
509 
534 
559 
585 
610 
635 
660 
686 
711 
736 
762 
787 
812 
837 
863 
888 
913 
938 
964 
989 
014 
040 
065 
090 

115 
141 
166 
191 
216 
242 
267 
292 
318 
343 



99 



99 



00 



Log. 



393 
419 
444 
469 
494 
520 
545 
570 
595 
621 
646 
67l 
697 
722 
747 
772 
798 
823 
848 
873 
899 
924 
949 
974 
000 
Cot, 



0.01 263 
0.01 238 
0.01 213 
0.01 187 
0.01 162 



0.01 137 
0.01 112 
0.01 086 
0.01 061 
Q.Ol 036 



0.01 010 
0.00 985 
0.00 960 
0.00 935 
0.00 909 



0.00 884 
0.00 859 
0.00 834 
0.00 808 
0.00 783 



0.01 516 
0.01491 
0.01465 
0.01 440 
0.01 415 



0.01 390 
0.01 364 
0.01 339 
0.01 314 
0.01 289 



0-00 758 
0.00 733 
0.00 707 
0.00 682 
0.00 657 



0.00 631 
0.00 606 
0-00 581 
0.00 556 
0-00 530 



0.00 505 
0.00 480 
0.00 455 
0-00 429 
0-00 404 



0-00 379 
0-00 353 
0.00 328 
0.00 303 
0-00 278 



0.00 252 
0.00 227 
0.00 202 
0.00 177 
0.00 151 



0-00 126 
0-00 101 
0-00 076 
0-00 050 
0-00 025 



n ■ no OOP 
Log. Tan, 



85 693 
85 681 
85 669 
85 657 
85 644 



85 632 
85 620 
85 608 
85 595 
85 583 



85 571 
85 559 
85 546 
85 534 
85 522 



85 509 
85 497 
85 485 
85 472 
85 460 



85 448 

85 435 
85 423 
85 411 
85 398 



85 386 
85 374 
85 36l 
85 349 
85 336 



85 324 
85 312 
85 299 
85 287 
85 274 



85 262 
85 249 
85 237 
85 224 
85 212 



85 199 
85 187 
85 174 
85 162 
85 149 



85 137 
85 124 
85 112 
85 099 
85 087 



85 074 
85 062 
85 049 
85 037 
85 024 



85 Oil 
84 999 
84 986 
84 974 
84 961 



84 948 
Log. Sin. 



P. p. 





25 


35 


6 


2-5 


2.5 


7 


3 





2.9 


8 


3 


4 


3.3 


9 


3 


8 


37 


10 


4 


2 


4.1 


20 


8 


5 


8.3 


30 


12 


7 


12.5 


40 


17 





16.6 


50 


21 


2 


20.^ 



13 



1 


3 


1 


6 


1 


8 


2 





2 


2 


4 


5 


6 


7 


9 





11 


2 



13 

1.3 
1.5 
1.7 
l.P 
2.1 
4.3 
6.5 
8.6 
10.8 





1^. 


6 


1.2| 


7 


1 


4 


8 


1 


6 


9 


1 


9 


IC 


2 


1 


20 


4 


] 


30 


6 


? 


40 


8 


3 


50 


10 


4 



12 
1-2 
1.4 
1.6 
1.8 
2-0 
4.0 
6.0 
8-0 
10.0 



P. P. 



134^ 



64^ 



46^ 



TABLE VIII.—LOGARITHMIC VERSED SINES AND EXTERNAL 
0° SECANTS. 1° 



' Log. Vers. 



2.62642 
3. 22848 
3-58066 
3. 83054 



4-02436 
.18272 
.31662 
.43260 
.53490 



4-62642 
. 70920 
•78478 
.85431 
.91868 



4-97860 

5-03466 

-08732 

.13696 

-18393 



5-22848 
.27086 
.31126 
.34987 
.38684 



5.42230 
.45636 
.48915 
.52073 
.55121 



5-58066 
.60914 
.63672 
.66344 
-68937 



5-71455 
.73902 
.76282 
.78598 
.80854 



5-83053 
•85198 
•87291 
•89335 
.91332 



5.93284 

.95193 

.97061 

5-98890 

6-00680 



6-02435 
.04155 
.05842 
.07496 
-09120 



6-10714 
-12279 
-13816 
-15327 
-16811 



618271 



D 



60206 
35218 
24987 
19382 
15836 
13389 
11598 
10230 
9151 
8278 
7558 
6953 
6437 
5992 
5605 
5266 
4964 
4696 
4455 
4238 
4040 
3861 
3697 
3545 
3406 
3278 
3158 
3048 
2944 
2848 
2757 
2672 
2593 
2518 
2447 
2379 
2316 
2256 
2199 
2145 
2093 
2044 
1996 
1952 
1909 
1868 
1829 
1790 
1755 
1720 
1686 
1654 
1623 
1594 
1565 
1537 
1511 
1484 
1460 



Log. Exsec. 





— CO 


2 
3 
3 
3 


62642 
22848 
58066 
83054 



4-02436 
.18272 
.31662 
.43260 
.53491 



4-62642 
.70921 
.78478 
.85431 
.91868 



4.97861 

5.03466 

•08732 

.13697 

.18393 



5-22849 
.27087 
.31127 
.34988 
•38685 



5-42231 
.45638 
•48916 
.52075 
.55123 



5.58068 
.60916 
•63674 
•66346 
•68940 



5 • 71457 
•73904 
•76284 
•78601 
-8085 7 



5-83056 
.85201 
.87295 
.89338 
•91335 



5-93288 

-95197 

•97065 

5-98894 

600685 



6-02440 
-04160 
•05847 
•07501 
-09125 



6-10719 
.12284 
-13822 
-15333 
-1 6818 

6 18278 



60206 
35218 
24987 
19382 
15836 
13389 
11598 
10230 

9151 
8279 
7557 
6952 
6437 
5993 
5605 
5266 
4964 
4696 
4456 
4238 
4040 
386l 
3697 
3545 
3407 
3278 
3159 
3048 
2945 
2848 
2758 
2672 
2593 

2517 
2447 
2380 
2316 
2256 
2199 
2145 
2093 
2043 
1997 
1952 
1909 
1868 
1829 
1791 

1755 
1720 
1687 
1654 
1623- 
1594 
1565 
1537 
1511 
1485 
1460 



' Log. Vers, 2> ILog. Exsec. 1> 



Log. Vers. 



6-18271 
-19707 
-21119 
.22509 
-23877 



6-25223 
-26549 
•27856 
-29142 
-30410 



6-31660 
•32892 
•34107 
•35305 
-36487 



6-37653 
•38803 
•39938 
•41059 
■42165 



6-43258 
-44337 
•45403 
.46455 
-47496 



-48524 
.49539 
•50544 
•51536 
•52518 



.53488 
• 54448 
•55397 
•56336 
-57265 



6 •58184 
•59093 
•59993 
•60884 
.61766_ 



6-62639 
•63503 
•64359 
•65206 
•66045 



6^66876 
•67700 
•68515 
•69323 
-70124 



6-70917 
-71703 
-72482 
-73254 
-74019 



6-74777 
-75529 
-76275 
-77014 
-77747 



6-78474 



Log. Vers. 



J> 



1435 

1412 

1389 

1368 

1346 

1326 

1306 

1286 

1268 

1250 

1232 

1214 

1198 

1182 

1166 

1150 

1135 

1121 

1106 

1093 

1078 

1066 

1052 

1040 

1028 

1016 

1004 

992 

981 

970 

960 

949 

939 

929 

919 

909 

900 

891 

882 

872 

864 

855 

847 

839 

831 
823 
815 
808 
80a 

793 
786 
779 
772 
765 
758 
752 
745 
739 
733 
726 



Log. Exsec, 



I> 



18278 
19714 
21126 
22516 
23884 



25231 
26557 
27864 
29151 
30419 



31669 
32901 
34116 
35315 
36497 



37663 
38814 
39949 
41070 
42177 



43270 
44349 
45415 
46468 
47509 



48537 
49553 
50557 
51550 
52532 



53503 
54463 
55413 
56352 
57281 



58201 
59110 
60011 
60902 
61784 



62657 
63522 
643 7B 
65226 
66065 



66897 
67720 
68536 
69345 
70145 
70939 
71725 
72505 
73277 
74043 



74802 
75554 
76300 
77040 
77773 



6 78500 



Log. Exsec, 



650 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL 
3° SECANTS. 3** 



' Log. Vers. 



D 



Log. Exsec. 



Log. Vers, 



Log. Exsec. 



6-78474 
.79195 
.79909 
.80618 
.81322 



5 


6.82019 


6 


.82711 


7 


.83398 


8 


.84079 


9 


.84755 


10 


6.85425 


11 


.86091 


12 


.86751 


13 


.87407 


14 


.88057 


15 


6-88703 


16 


.89344 


17 


.89980 


18 


.90612 


19 


.91239 


30 


6.91862 


21 


.92480 


22 


.93093 


23 


.93703 


24 


.94308 



6.94909 
.95506 
.96099 
.96688 
.97272 



6.97853 

.98430 

.99004 

6-99573 

7-00139 



.00701 
.01259 
.01814 
.02366 
•02914 



-03458 
•03999 
.04537 
-05071 
-05603 



-06130 
.06655 
.07177 
-07695 
-08211 



-08723 
.09232 
.09739 
.10242 
.10743 



.11240 
.11735 
.12227 
.12716 
-13203 



713687 
Log. Vers. 



721 
714 
709 
703 
697 
692 
686 
681 
676 
670 
665 
660 
655 
650 
646 
641 
636 
631 
627 
622 
618 
613 
609 
605 

601 
597 
592 
589 
584 

581 

577 
573 
569 
535 

562 
558 
555 
55l 
548 

544 
541 
537 
534 
53l 

527 
525 
52T 
518 
515 
51^ 
509 
506 
503 
500 

497 
495 
492 
489 
486 
484 



6.78500 
.79221 
.79937 
.80646 
.81350 



6.82048 
.82740 
.83427 
.84109 
.84785 



6.85457 
.86123 
.86783 
.87439 
.88090 



6.88737 
•89378 
.90015 
.90647 
.91275 



6-91898 
.92516, 
•93131 
•93741 
-94346 



6-94948 
.95545 
•96139 
•96728 
.97313 



6.97895 

.98472 

.99046 

6-99616 

7-00182 



7-00745 
.01304 
.01860 
.02412 
■02960 



7-03505 
.04047 
•04585 
•05120 
-05652 



7-06180 
.06706 
•07228 
.07747 
.08263 



7-08776 
.09286 
•09793 
•10297 
-10798 



7-11297 
•11792 
.12285 
•12775 
-13262 



7-1374fi 
Log. Exsec 



721 
715 
709 
703 
698 
692 
687 
682 
676 

67l 
666 
660 
656 
651 

646 
641 
636 
632 
628 
623 
618 
614 
610 
605 
601 
597 
593 
58? 
585 
581 
577 
574 
570 
566 

563 
55? 
555 
552 
548 

545 
54l 
538 
535 
53l 

528 
525 
522 
519 
516 
513 
509 
507 
503 
501 
498 
495 
493 
490 
487 
484 

n 



7-13687 
•14168 
•14646 
.15122 
•15595 



7.16066 
.16534 
.17000 
.17463 
•17923 



7-18382 
.18837 
.19291 
.19742 
•20191 



7-20637 
.21081 
.21523 
.21963 
•22400 



722836 
.23269 
•23700 
•2412? 
•24555 



7.24980 
.25402 
•25823 
•26241 
.26658 



7.27072 
•27485 
.27895 
•28304 
-28711 



7-29116 
•29518 
•29919 
•3031? 
-30716 



7-31112 
•31505 
•31897 
•32288 
.32676 



7.33063 
.33448 
.33831 
•34213 
-34593 



7-34971 
•35348 
•35723 
•36097 
-36468 



7-3683? 
.37207 
.37574 
•37940 
38304 



7-38667 
Log. Vers. 



651 



481 
478 
475 
473 
470 
468 
466 
463 
460 

458 
455 
453 
451 
448 
446 
444 
442 
440 
437 
435 
433 
431 
42? 
426 
424 
422 
420 
418 
416 
414 
412 
410 
40? 
406 
405 
402 
401 
39? 
397 
395 
393 
392 
390 
388 
386 
385 
383 
382 
380 

378 
377 
375 
373 
37l 
370 
368 
367 
366 
364 
362 



7-13746 
•14228 
•14707 
•15183 
•15657 



7.16129 
•16598 
•17064 
•17528 
.17989 



7.18448 
•18905 
•19359 
•19811 
•20260 



7.20707 
.21152 
.21595 
.22035 
■22473 



7.2290? 
.23343 
•23775 
•24204 
.24632 



7.25057 
.25480 
•25902 
•26321 
•26738 



7.27153 
.27567 
.27978 
•28387 
.28795 



7-29200 
.29604 
.30006 
.30406 
•30804 



7-31201 
.31595 
.31988 
•3237? 
-32768 



733156 
.33542 
.33926 
.3430? 
.34689 



7-3506? 
.35446 
.35822 
.36196 
•36569 



7-36940 
.37310 
•37678 
•38044 
•38409 



7-38773 
Log. Exsec 



TABLE VIII.— LOGARITHMIC VERSED SINES ANu EXTERNAL SECANTS. 
4° 5° 



Lg. Vers. J> Log.^Exs. 2> Lg. Vers. I> Log. Exs, 




42211 
42557 
42903 
43246 
43589 



43930 
44270 
44608 
44946 
45281 



45616 
45949 
46281 
46612 
46941 



47270 
47597 
47922 
48247 
48570 



48892 
49213 
49533 
49852 
50169 



50485 
50800 
51114 
51427 
51739 



52050 
52359 
52667 
52975 
53281 



53586 
53890 
54193 
54495 
54796 



55096 
55395 
55692 
55989 
56285 



56580 
56873 
57166 
57458 
57749 



58039 



Lg. Vers, 



361 
359 
358 
356 
355 
353 
352 
350 
349 
348 
346 
345 
343 
342 
341 
339 
338 
337 
335 
334 
333 
332 
330 
329 
328 
327 
325 
324 
323 
322 
321 
320 
318 
317 
316 
315 
314 
313 
311 
311 
309 
308 
307 
306 

305 
304 
303 
302 
300 
300 
299 
297 
297 
295 
295 
293 
293 
292 
290 
290 

Id 



38773 
39134 
39495 
39854 
40211 



40567 
40922 
41275 
41627 
41977 



42326 
42673 
43019 
43364 
43708 



44050 
44390 
44730 
45068 
45405 



45740 
46075 
46407 
46739 
47070 



47399 
47727 
48054 
48379 
48703 



49026 
49348 
49669 
49989 
50307 



50624 
50941 
51256 
51569 
51882 



52194 
52504 
52814 
53122 
53429 



53735 
54041 
54345 
54648 
54950 



55251 
55550 
55849 
56147 
56444 



56740 
57035 
57329 
57621 
57913 



7-58204 
Log.Exs 



361 
360 
359 
357 
356 
354 
353 
352 
350 
349 
347 
346 
345 
343 
342 
340 
339 
338 
337 
335 
334 
332 
332 
330 
329 
328 
327 
325 
324 

323 
322 
321 
319 
318 
317 
316 
315 
313 
313 

311 
310 
309 
308 
307 
306 
305 
304 
303 
302 

301 
299 
299 
298 
296 
296 
295 
294 
292 
292 
291 



7-58039 
.58328 
.58615 
.58902 
.59188 



7.59473 
.59758 
.60041 
•60323 
•60604 



7.60885 
61164 
61443 
61721 
61998 



7.62274 
•62549 
•62823 
•63096 
•63369 



7 •63641 
•63911 
•64181 
•64451 
•64719 



7 •64986 
•65253 
•65519 
•65784 
•66048 



7.66311 
.66574 
•66836 
.67097 
•67357 



7-67617 
•67875 
•68133 
•68390 
•68647 



7-68902 
•69157 
•69411 
•69665 
-69917 



7-70169 
•70421 
•70671 
•70921 
•71170 



7.71418 
•71666 
•71913 
•72159 
■ 72404 



7-72649 
-72893 
.73137 
.73379 
•73621 



7-73863 
Lg. Vers. 



289 
287 
287 
286 
285 
284 
283 
282 
281 
280 
279 
279 
277 
277 
276 
275 
274 
273 
272 

272 
270 
270 
269 
268 

267 
266 
266 
265 
264 
263 
263 
26i 
261 
260 

259 
258 
258 
257 
256 
255 
255 
254 
253 
252 
252 
251 
250 
250 
249 
248 
247 
247 
246 
245 
245 
244 
243 
242 
242 
241 



58204 
58494 
58783 
59071 
59358 



59645 
59930 
60214 
60498 
60780 



61062 
61342 
61622 
61901 
62179 



62456 
62733 
63008 
63282 
63556 



63829 
64101 
64372 
64643 
64912 



65181 
65449 
65716 
65982 
66247 



66512 
66776 
67039 
67301 
67562 



67823 
68083 
68342 
68601 
68858 



69115 
69371 
69627 
69881 
70135 



70388 
70641 
70893 
71144 
71394 



71644 
71892 
72141 
72388 
72635 



72881 
73126 
73371 
73615 
73859 



74101 



Log.Exs. 



2> 



290 
289 
288 

287 
286 
285 

284 
283 
282 

281 
280 
280 
279 
278 

277 
276 
275 
274 
274 
273 
272 
271 
270 
269 
269 
268 
267 
266 
265 
264 
264 
263 
262 
261 
261 
260 
259 
258 
257 
257 
256 
255 
254 
254 

253 
252 
252 
251 
250 
250 
248 
248 
247 
246 

246 
245 
245 
244 
243 
242 



P.P. 





360 


350 


340 


6 


36.0 


35-0 


34-0 


7 


42 





40 


8 


39 


6 


8 


48 





46 


6 


45 


3 


9 


54 





51 


5 


51 





10 


60 





58 


3 


56 


6 


20 


120 





116 


6 


113 


3 


30 


180 





175 





170 





40 


240 





233 


3 


226 


6 


50 


300 





291 


6 


283 


3 





330 


330 


310 


6 


33^0 


32^0 


31^0 


7 


38^5 


37^3 


36 


1 


8 


44^0 


42^6 


41 


3 


9 


49^5 


48^0 


46 


5 


10 


55-0 


53-3 


51 


g 


20 


110-0 


106-6 


103 


3 


30 


165-0 


160-0 


155 





40 


220^0 


213-3 


206 


6 


50 


275.0 


266-6 


258 


3 





300 


390 


6 


30^0 


29^0 


7 


35 





33 


g 


8 


40 





38 


6 


9 


45 





43 


5 


10 


50 





48 


3 


20 


100 





96 


5 


30 


150 





145 





40 


200 





193 


3 


50 


250 





241 


6 





270 


360 


35( 


6 


27^0 


26-0 


25^ 


7 


31^5 


30 


3 


29 • 


8 


36^0 


34 


6 


33 • 


9 


40^5 


39 





37^ 


10 


45^0 


43 


3 


41 • 


20 


90-0 


86 


6 


83- 


30 


135-0 


130 





125- 


40 


180-0 


173 


3 


166- 


50 


225-0 


216 


6 


208- 



380 

28^0 

32^6 

37-3 

42^0 

46^6 

93-3 

140^0 

186^6 

233 • 3 



6 

7 

8 

9 

10 

20 

30 

40 

50 



340 

24^0 

28^0 

32^0 

36^0 

40-0 

80-0 

120^0 

160-0 

200-0 



330 

23-0 

26 

30 



38 

78 

115 

153 

191 



330 

22.0 
25 



29 

33 

36 

73 

110 

146 

183 





310 


300 


19 





6 


21-0 


20-0 


19-0 


7 


24 


5 


23 


3 


22 


1 


8 


28 





26 


6 


25 


3 


9 


31 


5 


30 





28 


5 


10 


35 





33 


3 


31 


6 


20 


70 





66 


6 


63 


3 


30 


105 





100 





95 





40 


140 





133 


3 


126 


6 


50 


175 





166 


6 


158 


3 



P. p. 



652 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



10 

112 
cl4 



15 



19 



i20 
[21 
522 
£23 

324 

'25 
^-26 
y27 
828 
^29_ 
,30 
31 
32 
33 
|3£ 
135 
[36 
f37 
38 
S39_. 

t^ 

842 

3 

ke 
F 



50 

PI 
,52 

;3 

14 
5 
6 
7 
8 
S9 

^0 



Lg.Vers 



7-73863 
.74104 
. 74344 
.74583 
.74822 



7.75060 
.75297 
.75534 
.75770 
■76006 



7-76240 
.76475 
.76708 
•76941 
.77173 



7-77405 
.77636 
• 77867 
.78097 
-78326 



7-78554 
.78783 
.79010 
.79237 
-79463 



7.79689 
.79914 
.80138 
•80362 
-80586 



7-80808 
•81031 
•81252 
•81473 
-81694 



7-81914 
82133 
82352 
82570 
82788 





7-87238 
Lg. Vers 



l> Log.Exs 



241 
240 
239 
239 

238 

237 
236 
236 
235 
234 
234 
233 
233 
232 
232 
231 
230 
230 
229 

228 
228 
227 
227 
226 
225 
225 
224 
224 
223 
222 
222 
221 
221 
220 
220 
219 
219 
218 
217 
217 
217 
216 
215 
215 
214 
214 
213 
213 
212 
212 
211 
211 
210 
210 
209 
209 
208 
208 
207 
206 
■^ 





74101 
74343 
74585 
74826 
75066 



77664 
77897 
78128 
78360 
78590 



81088 
81312 
81535 
81758 
81980 



82201 
82422 
82642 
82862 
83081 



83300 
83518 
83735 
83952 
84169 



84385 
84600 
84815 
85030 
85243 



85457 
85670 
85882 
86094 
86305 



86516 
86726 
86936 
87146 
87354 



7-87563 
Log.Exs, 



Di Lg.Vers, 



242 
241 
241 
240 
239 
239 
238 
237 
237 
236 
235 
235 
234 
233 
233 
232 
231 
231 
230 
230 
229 
229 
228 
228 
227 
226 
226 
225 
225 
224 
224 
223 
222 
222 
221 
221 
220 
219 
219 
219 
218 
217 
217 
216 
216 
215 
215 
214 
213 
213 
213 
212 
211 
211 

211 
210 
210 
209 
208 
208 



7-87238 
• 87444 
.87650 
.87855 
-88060 



7-88264 
.88468 
.88672 
.88875 
.89077 



7.89279 
.89481 



.89882 
.90082 



7.90282 
.90481 
.90680 
.90878 
•91076 



7.91273 
.91470 
.91667 
.91863 
.92058 



7.92253 
.92448 
.92642 
•92836 
.93029 



7.93222 
.93415 
.93607 
.93799 
.93990 



7.94181 
.94371 
.94561 
.94751 
-94940 



7.95129 
.95317 
.95505 
.95693 
.95880 



7.96066 
.96253 
.96439 
.96624 
-96809 



7.96994 
.97178 
.97362 
.97546 
.97729 



7-97912 
.98094 
.98276 
.98458 
.98639 



7-98820 
Lg. Vers 



D Log.Exs. D 



206 
205 
2u5 
204 
204 
204 
203 
203 
202 

202 
201 
201 
200 
200 
199 
199 
198 
198 
197 
197 
197 
196 
196 
195 
195 
195 
194 
194 
193 
193 
192 
192 
191 

190 
190 
190 
189 
189 
189 
188 
187 
188 
187 

186 
186 
186 
185 
185 
184 
184 
184 
183 
183 
183 
182 
182 
182 
181 
181 



7-87563 
87771 
87978 
88185 
88391 



88597 
88803 
89008 
89212 
89416 



89620 
89823 
90025 
90228 
90429 



90630 
90831 
91032 
91231 
91431 



91630 
91828 
92027 
92224 
92421 



92618 
92815 
93010 
93206 
93401 



93596 
93790 
93984 
94177 
94370 



94562 
94754 
94946 
95137 
95328 



95519 
95709 
95898 
96088 
96276 



96465 
96653 
96841 
97028 
97215 



97401 
97587 
97773 
97958 
98143 



98327 
98512 
98695 
98879 
99062 



99244 



Log.Exs. 



208 
207 
207 
206 
206 
205 
205 
204 
204 
203 
203 
202 
202 
201 
201 
201 
200 
199 
199 
199 
198 
198 
197 
197 
197 
196 
195 
195 
195 
195 
194 
194 
193 
193 
192 
192 
192 
191 
191 
190 
190 
189 
189 
188 
188 
188 
188 
187 
187 
186 
186 
185 
185 
184 
184 
184 
183 
183 
183 
182 



O 

1 
2 
3 
4 

5 
6 
7 

8 

9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

30 

21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39_ 

40 

41 
42 
43 
il 
45 
46 
47 
48 
41 
50 
51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



P.P. 



180 9_ 



6 


18-0 


0-9 


7 


21-0 


1-1 


8 


24-0 


1.2 


9 


27-0 


1-4 


10 


30-0 


1-6 


20 


60-0 


3-1 


30 


90.0 


4-7 


40 


120-0 


6-3 


50 


150-0 


7-9 





8 


8 


6 


0-8 


0-8 


7 


1.0 


0-9 


8 


1.1 


1-0 


9 


1.3 


1.2 


10 


1.4 


1-3 


20 


2-8 


2-6 


30 


4-2 


4-0 


40 


5-6 


5-3 


50 


7.1 


6^6 





7 


6 


6 


0^7 


0.61 


7 


0-8 





7 


8 


0-9 





8 


9 


1-0 


1 





10 


1-1 


1 


1 


20 


2-3 


2 




30 


3-5 


3 


2 


40 


4^6 


4 


3 


50 


5.8 


5 


4 



9 

0-9 
1-0 
1.2 
1.3 
1.5 
3.0 
4.5 
6.0 



7_ 
0.7 
0-9 
1.0 
1.1 
1.2 
2.5 
3.7 
5.0 



6 

0.6 
0-7 





5 


5 




4 


6 


0-5 


0-5 


0.-4 


7 


0-6 





6 





5 


8 


0.7 





6 





6 


9 


0.8 





7 





7 


10 


0.9 





8 





7 


20 


1.8 


1 


g 


1 


5 


30 


2.7 


3 


5 


2 


2 


40 


3.6 


3 


3 


3 





50 


4-6 


4 


1 


3 


7 



10 
20 
30 
40 
50 



3 

0-3 
0-4 
0-4 
0.5 
0-6 
1.1 
1.7 
2.3 
2.9 



3 

0-3 
0-3 
0-4 
0-4 
0-5 
1.0 
1.5 
2.0 
2.5 



3 2 

0-2|0.2 
0-3 
0-30 
0-4;0 
0-4,0 
0-810 
1-2 1 
1.6 1 
2.1*1 





1 


1 


6 


0-1 


0.11 


7 


0-2 





' 


8 


0.2 





1 


9 


0.2 





■ 


10 


0.2 







200.5 





V 


30'0.9 





5 


40J1.0 





6 


50 


1.2 





8 



o 

0-0 
00 
0-0 
0-1 
0-1 
O.I 
0-2 
0.3 
0.4 



P.P. 



653 



TABLE VIII.— -LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
8° 9° 



Lg. Vers. D Log.Exs. D Lg.Vers. jD Log.Exs. 2> 






7-98820 


1 


.99000 


2 


.99180 


3 


.99360 


4 


.99539 


5 


7.99718 


6 


7.99897 


7 


8.00075 


8 


.00253 


9 


.00431 



10 

11 

12 
13 
14 



30 

21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



8.00608 
.00784 
.00961 
.01137 
.01313 



8-01488 
.01663 
.01838 
.02012 
-02186 




8.04076 
.04246 
.04416 
.04585 
.04754 



40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

6C 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



8-04922 
.05090 
.05258 
.05426 
-05593 



8-05760 
.05926 
.06093 
.06259 
-06424 



45 


8-06589 


46 


.06754 


47 


.06919 


48 


.07083 


49 


-07247 


50 


8-07411 


51 


.07575 


52 


.07738 


53 


.07900 


54 


-08063 


55 


8-08225 


56 


.08387 


57 


.08549 


58 


.08710 


59 


-08871 



8-09031 
Lg, Vers 



180 ^ 

180 

179 

179 

179 

178 

178 

177 

178 

177 

176 

176 

176 

176 

175 

175 

175 

174 

174 

173 

173 

173 

172 

172 

172 

171 

171 

171 

170 

170 

170 

169 

169 

169 

168 

168 

168 

167 

167 

167 

166 

166 

166 

165 

165 

165 

165 

164 

164 

164 

163 

163 

162 

162 

162 

161 

162 

161 

161 

160 



99244 
99427 
99609 
99790 
99971 





01940 
02117 
02293 
02469 
02645 



04556 
0472? 
04899 
05070 
05241 



05411 
05581 
05751 
05921 
06090 



06259 
06427 
06595 
06763 
06931 



07098 
07265 
07431 
07598 
07'r64 



07929 
08095 
08260 
08424 
08589 



08753 
08917 
09081 
09244 
09407 



09569 
Log.Exs 



182 
182 
181 
181 
180 
180 
180 
180 
179 
179 
178 
178 
178 
177 
177 
177 
176 
176 
175 
175 
175 
175 
174 
174 
173 
173 
173 
173 
172 
172 
171 
171 
171 
170 
170 
170 
170 
169 
169 
169 
168 
168 
168 
167 
167 
167 
166 
166 
166 
165 
165 
165 
164 
164 
164 
163 
164 
16^ 
163 
162 



09031 
09192 
09352 
09512 
09671 



09830 
09989 
10148 
10306 
10464 



10622 
10779 
10936 
11093 
11250 



11406 
11562 
11718 
11873 
12029 



12184 
12338 
12492 
12647 
12800 



12954 
13107 
13260 
13413 
13565 



13717 
13869 
14021 
14172 
14323 



14474 
14625 
14775 
14925 
15075 



15225 
15374 
15523 
15672 
15820 



15968 
16116 
16264 
16412 
16559 



16706 
16852 
16999 
17145 
17291 



17437 
17582 
17728 
17873 
18017 



18162 
Lg.Vers. 



160 

160 

160 

159 

159 

159 

158 

158 

158 

157 

157 

157 

157 

156 

156 

156 

155 

155 

155 

155 

154 

154 

154 

153 

153 

153 

153 

152 

152 

152 

152 

151 

151 

151 

151 

150 

150 

150 

149 

150 

149 

149 

149 

148 

148 

148 

148 

147 

147 

147 

146 

146 

146 

146 

145 

145 

145 

145 

144 

144 

17 



8-09569 
.09732 
.09894 
.10056 
.10217 



8.10378 
.10539 
.10700 
.10860 
.11020 



8.11180 
.11340 
.11499 
.11658 
.11816 



8.11975 
.12133 
.12291 
.12448 
-12605 



8-12762 
.12919 
.13075 
.13232 
.13387 



8-13543 
.13698 
.13854 
.14008 
.14163 



8-14317 
.14471 
.14625 
.14778 
-14932 



8-15085 
•15237 
.15390 
.15542 
•15694 



8-15846 
.15997 
.16148 
.16299 
.16450 



8-16600 
.16750 
.16900 
.17050 
■17199 



8-17349 
.17497 
.17646 
.17795 
.17943 



8-18091 
.18238 
.18386 
.18533 
-18680 



818827 
Log.Exs. 



162 

162 

162 

161 

161 

161 

160 

160 

160 

160 

159 

159 

159 

158 

158 

158 

158 

157 

157 

157 

157 

156 

156 

155 

156 

155 

155 

154 

154 

154 

154 

153 

153 

153 

153 

152 

152 

152 

152 

152 

15l 

151 

151 

150 

150 

150 

150 

149 

149 

149 

148 

149 

148 

148 

148 

147 

147 

147 

147 

146 



P.P. 



25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
3t 
60 



180 
18-0 
21-0 
24-0 
27-0 
30-0 
60-0 
90-0 



170 

17-0 
19-8 
22-6 
25-5 
28-3 
56-6 
85-0 



120-0 113-3 
150.01141.6 



16<^ 

16. ( 

I8.e 

2\.l 
24 -C 
26-e 
53-£ 
80 -C 
106- 1 
133.2 



150 



6 

7 

8 

9 

10 

20 

30 

40 

50 



15 





17 


5 


20 





22 


5 


25 


C 


50 





75 





100 





125 






0-9 


0-9 


0- 


1-1 


1-0 


1- 


1-2 


1-2 


1- 


1-4 


1-3 


1- 


1-6 


1-5 


1- 


3-1 


3-0 


2- 


4-7 


4-5 


4. 


6-3 


6-0 


5 


7.9 


7.5 


7 



8 




7 


0-8 


0-7 





9 


0-9 


1 





1-0 


1 


2 


1-1 


1 


3 


1-2 


2 


6 


2-5 


4 





3-7 


5 


3 


5-0 


6 


6 


6-2 



6 
7 
8 
9 

10 
20 
30 
4C 
50 



6 

0.6 
0-7 
0-8 
1-0 
1-1 
2-1 
3-2 
4-3 
5-4 



140 

14-0 
16-3 
18-6 
21-0 
23-3 
46-6 
70-0 
93-3 
116-6 

) 8 




1 
3 

i 

i 

6 
1 

7 
0-7 
0.8 
0.9 
1.0 

l-I 
2.3 
3-5 
4.6 
5-8 



6 

0-6 



II 



5 5 

0-510-5 

60 
7.10 

8 

9 

11 
7 2 
6 3 



P.P. 



654 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
10° 11° 



O 

1 
2 
3 

5 
6 
7 
8 

10 

11 
12 
13 
14 



Lg. Vers. I> Log.Exs. T> Lg.Vers. D Log.Exs 



8-18162 
.18306 
.18450 
.18594 
.18738 



15 
16 
17 
18 

ii 
20 

21 
22 
23 
21 
25 
26 
27 
28 
29^ 

30 

31 
32 
33 
3^ 

35 
36 
37 
38 
39 



8.21698 
•21837 
.21975 
.22113 
•22251 



40 

41 
42 
43 
4£ 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 

60 



8.18881 
.19024 
.19167 
.19309 
.19452 



8.19594 
.19736 
.19878 
.20019 
•20160 



8-20301 
•20442 
.20582 
.20723 
•20863 



8.21003 
.21142 
.21282 
.21421 
-21560 



•22389 
-22526 
.22663 
-22800 
-22937 



-23073 
-23209 
-23346 
-23481 
23617 



8-23752 
-23888 
-24023 
-24158 
•24292 



8-24426 
-24561 
-24695 
-24828 
•24962 



8-25095 
-25228 
-25361 
-25494 
-25627 



8-25759 
-25891 
-26023 
-26155 
.26286 



3-28417 



Lg. Vers, 



144 
144 
144 
143 
143 
143 
142 
142 
142 
142 
142 
142 
141 
14l 
141 
140 
140 
140 
140 
140 
139 
139 
139 
139 
138 
138 
138 
138 
137 
138 
137 
137 
136 
137 
136 
136 
136 
135 
136 
135 
135 
135 
135 
134 
134 
134 
134 
133 
133 

133 
133 
133 
132 
133 
132 
132 
132 
132 
131 
131 



18827 
18973 
19120 
19266 
19411 



19557 
19702 
19847 
19992 
20137 



20281 
20425 
20569 
20713 
20857 



21000 
21143 
21286 
21428 
21571 



21713 
21855 
21996 
22138 
22279 



22420 
22561 
22701 
22842 
22982 



23122 
23262 
23401 
23540 
23679 



23818 
23957 
24095 
24234 
24372 



24509 
24647 
24784 
24922 
25059 



25195 
25332 
25468 
25604 
25740 



25876 
26012 
26147 
26282 
26417 



26552 
26686 
26821 
26955 
27089 



8-27223 



146 
146 
146 
145 
145 
145 
145 
145 
144 
144 
144 
144 
144 
143 
143 
143 
143 
142 
142 
142 
142 
141 
141 
141 
141 
140 
140 
140 
140 
140 
140 
139 
139 
139 
139 
138 
138 
138 
138 

137 
138 
137 
137 
137 
136 
136 
136 
136 
136 
136 
135 
135 
135 
135 
134 
134 
134 
134 
134 
134 



Log.Exs 



8-26417 
.26548 
•26679 
•26810 
-26941 



Z> 



8-27071 
.27201 
•27331 
•27461 
.27590 



8.27719 
.27849 
.27977 
•28106 
.28235 



8.28363 
•28491 
•28619 
•28747 
.28875 



8.29002 
•29129 
.29256 
.29383 
.29510 



8.29636 
.29763 
.29889 
.30015 
.30140 



8.30266 
•30391 
•30516 
•30642 
.30766 



8.30891 

.31015 

.31140 

31264 

-31388 



8-31511 
31635 
.31758 
.31882 
.32005 



8-32128 
-32250 
-32373 
.32495 
-32617 



8-32739 
-32861 
-32983 
-33104 
-33225 



8-33347 
33468 
33588 
33709 
33829 



8-33950 
Lg. Vers 



131 

131 

131 

130 

130 

130 

130 

130 

129 

129 

129 

128 

129 

128 

128 

128 

128 

128 

127 

127 

127 

127 

127 

126 

126 

126 

126 

126 

125 

125 

125 

125 

125 

-124 

124 

124 

124 

124 

124 

123 

124 

123 

123 

123 

123 
122 
122 
122 
122 

122 
122 
121 
121 
121 
12l 
121 
120 
120 
120 
120 



8-27223 
.27356 
.27490 
•27623 
-27756 



1-27889 
-28021 
.28153 
.28286 
.28418 



8.28550 
.28681 
•28813 
•28944 
•29075 



X> 



8-29206 
.29336 
.29467 
•29597 
.29727 



8.29857 
.29987 
.30117 
•30246 
.30375 



8.30504 
.30633 
•30762 
•30890 
-31019 



i- 31147 
.31275 
.31402 
•31530 
-31657 



•31785 
•31912 
•32039 
•32165 
.32292 



8-32418 
•32544 
•32670 
•32796 
.32922 



8.33047 
•33173 
.33298 
.33423 
- 33547 
-33672 
.33797 
.33921 
.34045 
^341_69 
-34293 
-34417 
- 34540 
-34663 
-34786 



8-34909 
Log.Exs 



133 
133 
133 
133 
133 
132 
132 
132 
132 
132 
131 
131 
131 
131 

131 
130 
130 
130 
130 
130 
130 
129 
129 
129 
129 
129 
128 
128 
128 
128 
128 
127 
127 
127 
127 
127 
127 
126 
126 
126 
126 
126 
126 
125 
125 
126 
125 
125 
124 
125 
124 
124 
124 
123 
124 
124 
123 
123 
123 
123 



o 

1 

2 
3 
4 
5 
6 
7 
8 
9 

10 

11 
12 
13 
Ik 
15 
16 
17 
18 
19 

30 

21 
22 
23 

25 
26 
27 
28 
29 

30 

31 
32 
33 

a4 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
ii 
50 
51 
52 
53 
11 
55 
56 
57 
58 
59. 
60 



P. P. 



130 



13 





15 


1 


17 


3 


19 


5 


21 


6 


43 


3 


65 





86 


6 


108 


3 





3 


4 


6 


0-4 


0-4 


7 


0-5 





4 


8 


0-6 





5 


9 


0-7 





6 


10 


0.7 







20 


1-5 


1 


3 


30 


2.2 


2 





40 


3.0 


2 


6 


50 


3.7 


3 


3 





3 


6 


0-3 


7 


0-3 


8 


0-4 


9 


0-4 


10 


0-5 


20 


1-0 


30 


1-5 


40 


2-0 


50 


2.5 



2 

0-2 



0-6 
1-0 
1-3 
1-6 



1 

0-1 
0.1 
0.1 
0-1 
0-1 
0-3 
0-5 
0-6 
0-8 



130 

12.0 
14.0 
16.0 
18.0 
20.0 
40.0 
60.0 
80.0 
100.0 



3 

0.3 
0.4 
0.4 
0.5 
0.6 
l-I 
1-7 
2.3 
2^9 



3 

0^2 
0-3 
0.3 
0.4 
0.4 
0-8 
1.2 
1.6 
2.1 



1 
O.I 
0.2 
0.2 
0.2 
0.2 
0.5 
0.7 
1-0 
1-2 



0_ 

0.0 
0-0 
0-0 
0-1 
0-1 
0-1 
0-2 
0^3 
0-4 



P. P. 



655 



lABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
13° 13° 



Lg. Vers 



8-33950 
34070 
34190 
34309 
34429 



34549 
34668 
34787 
34906 
35025 



35143 
35262 
35380 
35498 
35616 



35734 
35852 
35969 
36086 
36204 



36321 
36437 
36554 
36671 
36787 




38057 
38171 
38286 
38400 
38514 



38628 
38741 
38855 
38969 
39082 




39758 
39871 
39983 
40095 
40207 



40318 
40430 
40541 
40652 
40764 



8-40875 
Ig. Vers. 



-Z> Log.Exs 



120 
120 
119 
120 

119 
119 
119 
119 
119 
118 
118 
118 
118 
118 
117 
118 
117 
117 
117 
117 
116 
117 
116 
116 
116 
116 
116 
115 
115 
115 
115 
115 
115 
115 
114 
114 
114 
114 
114 
114 
113 
114 
113 
113 

113 
113 
113 
113 
112 

112 
112 
112 
112 
112 

111 
111 
111 
111 
111 

in 




34909 
35032 
35155 
35277 
35399 



35522 
35644 
35765 
35887 
36009 



36130 
36251 
36372 
36493 
36614 



36734 
36855 
36975 
37095 
37215 



3793T 
38050 
38169 
38287 
38406 
38524 
38642 
38760 
38878 
38995 

39113 
39230 
39347 
39464 
39581 



39698 
39814 
39931 
40047 
401 6P 




I> Log.Exs 



l> Lg. Vers. -^ Log.Exs 



123 
122 
122 
122 



120 
120 
120 
120 
120 



1 o o 



7> 



8-40875 
40985 
41096 
41206 
41317 



-41427 
.41537 
•41647 
.41757 
•41867 



-41976 
•42086 
•42195 
.42304 
.42413 



.42522 
.42630 
.42739 
•42847 
.42956 



.43064 
•43172 
•43280 
•43388 
•43495 
•43603 
•43710 
•43817 
•43924 
•44031 



.44138 
.44245 
•44351 
•44458 
.44564 
.44670 
.44776 
.44882 
•44988 
•45093 



•45199 
•45304 

-45409 
.45514 
•45619 



•45724 
•45829 
•4593^ 
•4603P 
J-m42 

- 46247 
-46351 
-46455 
•46558 
•46662 



-46766 
-46869 
.46972 
.47076 
-47179 



Lg. Vers. 



110 
110 
110 
110 
110 
110 
110 
109 
110 
109 
109 
109 
109 
109 

109 
108 
109 
108 
108 
108 
108 
108 
108 
107 
107 
107 
107 
107 
107 
106 
107 
106 
106 
106 

loe 

106 
105 
106 
105 
105 
105 
105 
105 
105 

105 
104 
105 
104 
104 

104 
104 
104 
103 
104 

103 
103 
103 
103 
103 
103 

IT 



•42569 
.42682 
.42795 
.42908 
-43021 



-43133 
.43246 
.43358 
•43470 
•43582 



•43694 
•43805 
•43917 
.44028 
-44139 



.46446 
.46555 
.46663 
.46771 
-46879 



-46987 
.47095 
.4720P 
-47310 
-47417 



-47525 
-47632 
.47739 
-47846 
-47953 



•48060 
-48166 
.48273 
.48379 
-48485 



8.48591 



Log.Exs. 



107 
106 
106 
106 
106 
106 

IT 



10 

11 

12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 

,24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
11 
35 
36 
37 
38 
39, 

40 

41 
42 
43 
44^ 

45 
46 
47 
48 
49^ 

50 

51 
52 
53 
5£ 
55 
56 
57 
58 
59_ 
P,0 



656 



P.P. 





130 


119 


6 


12-0 


11-9 


7 


14 





13 


9 


8 


16 





15 


8 


9 


18 





17 


8 


10 


20 





19 


8 


20 


40 





39 


6 


30 


60 





59 


5 


40 


80 





79 


3 


50 


100 





99 


1 



118 

11.8 
13.7 
15.7 
17.7 
19.6 
39.3 
59.0 
78.6 





117 116 


115 


6 


11.7 


11-6 


11.5 


7 


13 


6 


13 


5 


13.4 


8 


15 


6 


15 


4 


15-3 


9 


17 


5 


17 


4 


17-2 


10 


19 


5 


19 


3 


19-1 


20 


39 





38 


6 


38-3 , 


30 


58 


5 


58 





57-5 1 


40 


78 





77 


3 


76.6 1 


50 


97 


5 


96 


6 


95.8 1 





114 113 


113 f 


6 


11-4 


11.3 


11.2 ■ 


7 


13 


3 


13 


2 


13.0 


8 


15 


2 


15 





14.9 


9 


17 


1 


16 





16.8 


10 


19 





18 


8 


18.6 


20 


38 





37 


6 


37.3 


30 


57 





56 


5 


56.0 


40 


76 





75 




74.6 


50 


95 





94 


1 


93.3 w 



10 
20 
30 
40 
50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



111 110 109 



11.0 
12-8 
14-6 
16-5 
18-3 
36-6 
55-0 
73-3 
91-6 



10-9 
12-7 
14-5 
16.3 
18.1 
36.3 
54-5 
72-6 
90.8 



105 



e 

7 
8 
9 

10 
20 
30 
40 
50 



104 



•5 
-2 
.0 
• 7 
-5 
.0 
■ 5 
.0 

_5 

P. P. 



O 

0-0 
0.0 
0-0 
0.1 
0.1 
O.I 
0.2 
0.3 
0.4 



108 107 


106 


10-8 


10-7 


10-6 


12 


6 


12 


5 


12 


3 


14 


4 


14 


2 


14 


1 


16 


2 


16 





15 


9 ' 


18 





17 


8 


17 


6 


36 





35 


6 


35 


3 


54 





53 


5 


53 





72 





71 


3 


70 


6 


90 





89 


1 


88 


3 



ii 



TABLEVIII.— LOGARITHMIC VERSED SINES AND EXTERNAL. SEC ANTS. 
14° 15° 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

60 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Lg.Vers. I> Log.Exs. -Z> Lg.Vers, 



47282 
47384 
47487 
47590 
47692 



47795 
47897 
47999 
48101 
48203 



48304 
48406 
48507 
48609 
48710 



48811 
48912 
49013 
49114 
49215 



49315 
49415 
49516 
49616 
49716 



49816 
49916 
50015 
50115 
50215 
50314 
50413 
50512 
50611 
50710 



50809 
50908 
51006 
51105 
51203 



51301 
51399 
51497 
51595 
51693 



51791 
51888 
51986 
52083 
52180 




8.53242 



Lg, Vers 



102 
103 
102 
102 
102 
102 
102 
102 
102 

101 
101 
101 
101 
101 

101 

101 

101 

100 

101 

100 

100 

100 

100 

100 

100 

100 

99 

100 

99 

99 

99 

99 

99 

99 

98 

99 

98 

98 

98 

98 

98 

98 

98 

97 

98 

97 

97 

97 

97 

97 

97 

97 

96 

97 

96 

96 

96 

96 

96 

96 



8-49120 
.49225 
.49331 
.49436 
.49541 



8.49646 
.49750 
.49855 
.49960 
.50064 



8.50168 
.50273 
.50377 
.50481 
♦50585 



8-50688 
.50792 
.50896 
.50999 
.51102 



8-48591 
.48697 
.48803 
.48909 
-49014 




8.52231 
.52333 
.52435 
.52537 
.5263P 



8-52740 
.52841 
.52943 
.53044 
-53145 



8-53246 
.53347 
.53448 
.53548 
.53649 



8.53749 
.53850 
.53950 
.54050 
.54150 




Log. Exs 



106 
106 
105 
105 
105 
105 
105 
105 
105 

105 
104 
105 
104 
104 
104 
104 
104 
104 
104 
103 
104 
103 
103 
103 
103 
103 
103 
102 
103 
102 
102 
102 
102 
102 
102 
102 
102 
101 
101 
101 
101 
101 
101 
101 
101 
101 
101 
100 
100 
100 
100 
100 
100 
100 
100 
100 

99 
100 

99 



.53242 
.53338 
.53434 
.53530 
.53625 



-53721 
.53816 
.53911 
.54007 
-54102 



.54197 
.54291 
.54386 
.54481 
- 54575 
.54670 
.54764 
.54858 
.54952 
-55046 



.55140 
.55234 
.55328 
.55421 
.55515 



.55608 
.55701 
.55795 
.55888 
-55981 



-56074 
.56166 
.56259 
.56352 
-56444 



.56536 
.56629 
.56721 
.56813 
-56905 



.56997 
.57089 
.57180 
.57272 
-57363 



.57455 
.57546 
.57637 
.57728 
.57819 



-57910 
.58001 
.58092 
.58182 
.58273 



-58363 
.58453 
.58544 
-58634 
-58724 



8.58814 
Lg. Vers 



Log. Exs, 



-54748 
. 54847 
.54946 
.55045 
-55144 



-55243 
.55342 
-55441 
-55539 
-55638 



-55736 
-55834 
.55933 
.56031 
-56129 



-56226 
-56324 
.56422 
.56519 
■56617 



-56714 
.56812 
.56909 
.57006 
.57103 



-57200 
.57296 
.57393 
.57490 
,57586 



8-57682 
.57779 
.57875 
.57971 
-58067 



8-58163 
.58259 
.58354 
.58450 
•58546 



8-58641 
.58736 
•58832 
.58927 
-59022 



8-59117 
.59211 
.59306 
.59401 
-59495 



8-59590 
.59684 
.59779 
.59873 
-59967 



8-60061 
.60155 
-60249 
-60342 
•60436 



•80530 



Log. Exs 



2> 



98 



2> 



P.P. 



6 

7 
8 
9 
10 
20 
30 
40 
50 



103 

10.3 
12.0 
13.7 
15-4 
17-1 
34-3 
51-5 
68-6 
85-8 



103 101 



10.2 
11.9 
13.6 
15-3 
17-0 
34-0 
51-0 
68-0 
85-0 



6 

7 

8 

9 

10 

20 

30 

40 

50 



100 99 



10-0 


9-9 


11-6 


11-5 


13-3 


13.2 


15-0 


14-8 


16-6 


16-5 


33-3 


33-0 


50-0 


49.5 


66-6 


86.0 


83.3 


82.5 





97 


96 


6 


9.7 


9-6 


7 


11.3 


11-2 


8 


12-9 


12-8 


9 


14-5 


14-4 


10 


16-1 


16-0 


20 


32-3 


32-0 


30 


48-5 


48-0 


40 


64-6 


64-0 


50 


80.8 


80.0 



10.1 
11.8 
13-i 
15.1 
16-8 
33-6 
50-5 
67-3 
84.1 



98 

9-8 
11-4 
13-0 
14-7 
16-3 
32-6 
49.0 
65.3 
82.6 



95 

9-5 
11.1 
12-6 
14-2 
15-8 
31-6 
47-5 
63-3 
79.1 





94 


93 


93 


6 


9.4 


9-3 


9.2 


7 


LO-9 


L0.8 


10.7 


8 


L2-5 


L2-4 


12.2 


9 


L4-1 


13-9 


13-8 


10 


L5-6 


L5.5 


15.3 


20 


31-3 


31-0 


30-6 


30 


47-0 


46-5 


46-0 


40 


62-6 


32-0 


61-3 


50 


78-3 


77-5 


76-6 


91 90 


6 


9-] 


9.( 


0-0 


7 


10-6 


10. f 


0-(i 


8 


12-1 


12.( 


0-(i 


i 


13.6 


13-f 


0.1 


IC 


15-1 


15. ( 


0.1 
O.I 


2( 


30-3 


30-1 


3C 


45.E 


45-1 


0.2 


4( 


60.6 


60-1 


0.3 


5C 


75.8 


75. ( 


0.4 



P.P. 



657 



tABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
16^ 17° 



Lg.Vers. J> Log.Exs, 1> Lg.Vers. l> Log.Exs 



58814 
58904 
58993 
59083 
59173 



59262 
59351 
59441 
59530 
59619 



59708 
59797 
59886 
59974 
60063 



60152 
60240 
60328 
60417 
60505 



60593 
60681 
60769 
60857 
60944 



61032 
61119 
61207 
61294 
61381 



61469 
61556 
61643 
61730 
61816 



61903 
61990 
62076 
62163 
62249 



62336 
62422 
62508 
62594 
62680 



62766 
62852 
62937 
63023 
63108 



63194 
63279 
63364 
63449 
63534 



63619 
63704 
63789 
63874 
63959 



8-64043 
Lg. Vers, 



8.60530 
.60623 
.60716 
.60810 
.60903 



8.60996 
.61089 
.61182 
.61275 
.61368 



8.61460 
.61553 
.61645 
.61738 
.61830 



8.61922 
.62014 
.62106 
.62198 
•62290 



8.62382 
.62474 
.62565 
.62657 
.62748 



8.62840 
.62931 
.63022 
.63113 
.63204 



8.63295 
.63386 
.63477 
.63567 
.63658 



8.63748 
.63839 
.63929 
.64019 
.64109 



8.64199 
.64289 
.64379 
.64469 
.64559 



8.64649 
.64738 
.64828 
.64917 
.65006 



8.65096 
.65185 
.65274 
.65363 
.65452 



8.65541 
.65629 
.65718 
.65807 
.65895 



8-65984 
Log.Exs, 



89 



J> 



. 64043 
.64128 
.64212 
.64296 
.64381 



. 64465 
.64549 
.64633 
.64717 
.64801 



. 64884 
.64968 
.65052 
.65135 
.65218 



8. 



65302 
65385 
65468 
65551 
65634 



.65717 
.65800 
.65883 
.65965 
.66048 



.66131 
.66213 
.66295 
.66378 
.66460 



.66542 
.66624 
.66706 
.66788 
.66870 



.66951 
.67033 
.67115 
.67196 
.67277 



.67359 
.67440 
.67521 
.67602 
.67683 



.67764 
.67845 
.67926 
.68007 
.68087 



.68168 
.68248 
.68329 
.68409 
.68489 



.68569 
.68650 
.68730 
.68810 
.68889 



8-68969 
Lg. Vers. 



8.65984 
.66072 
.66160 
.66248 
.66336 



8.66425 
.6651^ 
.66600 
.66688 
.66776 



.66863 
.66951 
.67039 
.67126 
.67213 



8-67301 
.67368 
.67475 
.67562 
.67649 



.67736 
.67822 
.67909 
.67996 
.68082 



.68169 
.68255 
.68341 
.68428 
.68514 



8.68600 
.68686 
.68772 
.68858 
.68944 



8.69029 
.69115 
.69201 
.69286 
-69372 



.69457 
.69542 
.69627 
.69712 
.69798 



.69883 
.69967 
.70052 
.70137 

.70222 



8.70306 
.70391 
.70475 
.70560 
.70644 



8.70728 
.70813 
.70897 
.70981 
•71065 



8-71149 



Log.Exs. 



2> 



10 

11 
12 
13 
-li 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

25 
26 
27 
28 

29 

30 

31 
32 
33 
_34 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 
45 
46 
47 
48 
_49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
fiO 



P.P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



93 

9.3 
10^8 
12.4 
13.9 
15.5 
31.0 
46.5 
62.0 
77.5 



93 

9.2 
10^7 
12.2 
13.8 
15^3 
30 • 6 
46^0 
61^3 
76.6 



91 

9.1 
10.6 
12^1 
13^6 
15-1 
30-5 
45-5 
60.6 
75^8 





90 


89 


88 


6 


9^0 


8.9 


8.8 


7 


10^5 


10-4 


10 


2 


8 


12.0 


11.8 


11 


7 


9 


13.5 


13-3 


13 


2 


10 


15.0 


14-8 


14 




20 


30.0 


29.6 


29 


3 


30 


45.0 


44.5 


44 





40 


60.0 


59.3 


58 


6 


50 


75.0 


74.1 


73 


3 



7 
8 

9 
10 
20 
30 
40 
50 



6 

7 

8 

9 

10 

20 

30 

40 

50 



87 
8.7 
10. 1 
11.6 
13.6 
14.5 
29.0 
43.5 
58.0 
72.5 

84 
8.4 
9-8 
11.2 
12.6 
14.0 
28.0 
42.0 
56.0 
70.0 



86 

8.6 

10.0 
11.4 
12.9 
14-3 
28-6 
43-0 
57-3 
71.6 



85 
8-5 



83 

8 
9 

11 
12 
13 
27 
41 
55 
69 





81 


80 , 


6 


8.1 


8-0 


7 


9-4 


9 


3 


8 


10.8 


10 


6 


9 


12.1 


12 





10 


13-5 


13 


3 


20 


27.0 


26 


6 


30 


40.5 


40 





40 


54.0 


53 


3 


50 


67.5 


66 


6 



83 
8-2 
9-5 
10-9 
12-3 
13-6 
27.3 
41-0 
54-6 
68.3 

79 

7-9 
9-2 
10.5 
11.8 
13.1 
26-3 
39-5 
52.6 
65.8 



0.0 
0-0 
0.1 
0.1 
0-1 
0-2 
0-3 
0.4 



P. P. 



658 



• TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

IS'' 19° 



Lg.Vers. I> Log.Exs. J> Lg.Vers. J> Log.Exs. T> 



69049 
69129 
69208 
69288 



69367 
69446 
69526 
69605 
69684 



69763 
69842 
69921 
70000 
70079 

70157 
70236 
70314 
70393 
70471 



70550 
70628 
70706 
70784 
70862 



70940 
71018 
71096 
71174 
71251 



71329 
71406 
71484 
71561 
71639 



71716 
71793 
71870 
71947 
72024 



72101 
72178 
72255 
72331 
72408 



72485 
72561 
72637 
72714 
72790 



72866 
72942 
73018 
73094 
73170 



73246 
73322 
73398 
73473 
73549 



73625 
. Vers. 



8.71149 
.71232 
.71316 
.71400 
•71484 



8.71567 
.71651 
.71734 
.71817 
•71901 



8-71984 
•72067 
.72150 
•72233 
•72316 



8.72399 
•72481 
.72564 
.72647 
•72729 



8^72812 
•72894 
•72977 
.73059 
•73141 



•73223 
.73306 
.73388 
.73470 
•73551 



•73633 
•73715 
.73797 
.73878 
.73960 



. 74041 
.74123 
.74204 
.74286 
•74367 



8 . 74448 
.74529 
.74610 
.74691 
.74772 



8.74853 
.74934 
.75014 
.75095 
•75175 



8^75256 
.75336 
.75417 
.75497 
•75577 



8.75658 
•75738 
.75818 
.75898 
•75978 



8-76058 
Log.Exs. 



8-73625 
.73700 
.73775 
.73851 
.73926 



8.74001 
.74076 
.74151 
.74226 
•74301 



8^74376 
.74451 
.74526 
.74600 
•74675 



8.74749 
.74824 
.74898 
.74973 
.75047 



.75121 
.75195 
.75269 
.75343 
.75417 



.75491 
.75565 
.75639 
.75712 
.75786 



8.75860 
.75933 
.76006 
.76080 
.76153 



8.76226 
.76300 
.76373 
.76446 
•76519 



8 • 76592 
.76664 
.76737 
.76810 
.76883 



8.76955 
.77028 
.77100 
.77173 
.77245 



8.77317 
.77390 
.77462 
.77534 
.77606 



8.77678 
.77750 
.77822 
.77893 
.77965 



8.78037 
Lg, Vers, 



75 
75 
75 
75 
75 
75 
75 
75 
75 
75 
74 
75 
74 
74 
74 
74 
74 
74 
74 
74 
74 
74 
74 
74 
74 
73 
74 
73 
73 
74 
73 
73 
73 
^3 

73 
73 
73 
73 
73 
73 
72 
73 
72 
73 
72 
72 
72 
72 
72 
72 
72 
72 
72 
72 
72 
72 
72 
71 
72 

7l 



8.76058 
.76137 
.76217 
.76297 
•76376 



8.76456 
.76536 
.76615 
.76694 
•76774 



8 •76853 
.76932 
.77011 
.77090 
•77169 



8 • 77248 
•77327 
.77406 
.77485 
•77563 



8.77642 
•77720 
•77799 
.77877 
•77956 



8.78034 
.78112 
.78191 
.78269 
.78347 



8.78425 
.78503 
.78581 
.78659 
.78736 



8.78814 
.78892 
.78969 
.79047 
•79124 



8.79202 
.79279 
.79357 
.79434 
.79511 



8.79588 
.79665 
.79742 
.79819 
.79896 



8.79973 
.80050 
.80126 
.80203 
.80280 



8.80356 
.80433 
.80509 
.80586 
.80662 



8-80738 



Log.Exs, 



P.P. 



10 

11 
12 
13 
ii 
15 
16 
17 
18 
19 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

30 

31 
32 
33 

34 

35 
36 
37 
38 
39_ 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49_ 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59_ 
60 





84 


83 


6 


8.4 


8.3 


7 


9.8 


9.7 


8 


11.2 


11.0 


9 


12.6 


12.4 


10 


14^0 


13.8 


20 


28^0 


27-6 


30 


42.0 


41.5 


40 


56.0 


55.3 


50 


70.0 


69.1 



83 
8.2 
9.5 
10.9 
12^3 
13^6 
27.3 
41.0 
54.6 
68.3 



6 

7 
8 
9 
10 
20 
30 
40 
50 



81 


80 


8.1 


8.0 


9.4 


9.3 


10.8 


10.6 


12^1 


12.0 


13^5 


13.3 


27^0 


26.6 


40^5 


40.0 


54.0 


53.3 


67.5 


66-6 





78 


77 


6 


7^8 


7.7 


7 


9^1 


9.0 


8 


10^4 


10.2 


9 


11.7 


11.5 


10 


13.0 


12-8 


20 


26-0 


25-6 


30 


39.0 


38.5 


40 


52.0 


51.3 


50 


65.0 


64.1 



9 
10 
20 
30 
40 
50 



75 


74 


7.5 


7.4 


8.7 


8.6 


10.0 


9.8 


11.2 


11.1 


12.5 


12.3 


25.0 


24^6 


37.5 


37^0 


50.0 


49^3 


62.5 


61.6 



73 


71 


7.2 


7.1 


8.4 


8.3 


9^6 


9.4 


10^8 


10.6 


12^0 


11.6 


24^0 


23.6 


36^0 


35.5 


48^0 


47.3 


60^0 


59. ll 



76 

7.6 
8.8 

10. 1 
11.4 
12.6 
25.3 
38-0 
50^6 
63-3 



73 

7^3 
8^5 
9.7 

10-9 
12^1 
24^3 
• 5 
48^6 
60^8 



h 

0.1 
0.1 
O.T 
0.2 
0.3 
0.4 



P.P. 



659 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
20* 21° 



Lg.Vers. J> Log.Exs. 2> Lg.Vers. 1> Log.Exs, 




78037 
78108 
78180 
78251 
78323 



78394 
78466 
78537 
78608 
78679 



78750 
78821 
78892 
78963 
79034 



79458 
79528 
79598 
79669 
79739 



79809 
79879 
79949 
80019 
80089 



80159 
80229 
80299 
80369 
80438 




80855 
80924 
80993 
81063 
81132 



81201 
81270 
81339 
81407 
81476 



81545 
81614 
81682 
81751 
81819 




Lg.Vers, 



n 



80738 
80814 
80891 
80967 
81043 



81119 
81195 
81271 
81346 
81422 



81498 
81573 
81649 
81725 
81800 



81876 
81951 
82026 
82102 
82177 



82252 
82327 
82402 
H2477 
32552 



82627 
82702 
82776 
82851 
82926 



83000 
83075 
83149 
83224 
83298 



83373 
83447 
83521 
83595 
83670 

83744 
83818 
83892 
83966 



84113 

84187 

8426 

8433: 

84408 



84481 
84555 
8462§ 
84702 
84775 



84848 
84922 
84995 
85068 
85141 



8-85214 



Log.Exs, 



.82229 
.82297 
.82366 
.82434 
.82502 



•82569 
.82637 
.82705 
.82773 
.82841 



.82908 
.82976 
.83043 
.83111 
.83178 



.83246 
.83311 
.83381 
.8344' 
.83515 



.83582 
.83649 
.83716 
.83788 
■83850 



.83916 
.83983 
.84050 
.84117 
.84183 



8. 



84250 
84316 
84388 
84449 
84515 



•84582 
.84648 
.84714 
.84780 
• 84846 



.84912 
•84978 
.85044 
.851ir 
.8517i 



85242 
.85308 
.85373 
.85439 
.85505 



.855^0 
.85626 
. 85*701 

• ss-^ee 

.85832 



8. 



85897 
85962 
86027 
86092 
86158 



8.86223 



J^ Lg.Vers, 



.85214 
.85287 
.85360 
.85433 
.85506 



.85579 
.85651 
.85724 
.85797 
.85869 



.85942 
.86014 
.86087 
.86159 

.86231 



.86304 
.86376 
.86448 
.86520 
.86592 



.86664 
.86736 
.8680g 
.86880 
•86952 



8.87024 
87095 
87167 
87239 
87310 
8.87382 
.87453 
.87525 
.87596 
.87668 



8.87739 
.87810 
.87881 
.87953 
•88024 



8.88095 
.88166 
.88237 
.88308 
■88378 



8.88449 
.88520 
.88591 
.88661 
.88732 



8.88O03 
.888''3 
o 88944 
.89014 
■89085 



.89155 
.89225 
.89295 
.89366 
•89436 



8.895061 



Log.Exs. 





1 
2 
3 
A 

5 
6 
7 
8 

A 

10 

11 

12 
13 

li 
15 
16 
17 
18 
ii 
20 
21 
22 
23 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
M 
55 
56 
57 
58 
59 
60 



P.P. 





76 


75 


74 


6 


7.6 


7.5 


7.4 


7 


8.8 


8.7 


8.6 


8 


10.1 


10.0 


9.8 


9 


11.4 


11.2 


11.1 


10 


12.6 


12.5 


12. S 


20 


25.3 


25.0 


21.6 


30 


38.0 


37.5 


37.0 

49. § 


40 


50.6 


50.0 


50 


63.3 


62-5 


61.6 



6 

7 
8 
9 
10 
20 
30 
40 
50 



73 

7.S 
8.5 
9^7 
10.9 
12.1 
24-3 
36.5 
48.6 
60.8 



72 

7.2 

8.4 

9.6 

10.8 

12.0 

24.0 

36.0 

48.0 

60.0 





70 


69 


68 


6 


7.0 


6.9 


6.8 


7 


8.1 


8.0 


9^0 


8 


9-3 


9.2 


9 


10.5 


10.3 


10.2 


10 


11.6 


11.5 


11.3 


20 


23.3 


23.0 


22.6 


30 


35-0 


34.5 


34.0 


40 


46.6 


46.0 


45.3 


50 


58.3 


57.5 


56.6 





67 


66 


6 


6.7 


6.6 


7 


7.8 


7.7 


8 


8-9 


8-8 


9 


10.0 


9.9 


10 


11.1 


11.0 


20 


22.3 


22.0 


30 


33.5 


33.0 


40 


44.6 


44.0 


50 


55.8 


55.0 



65 

6.5 

7^6 

8.6 

9.7 

10.8 

21.6 

32.5 

43.3 

54.1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



P.P. 



660 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

33° 33° 



Lg.Vers. J> Log.Exs. 1> Lg.Vers. 2> Log.Exs. 1> 



86223 
86287 
86352 
86417 
86482 

86547 
86612 
86676 
86741 
86805 



86870 
86934 
86999 
87063 
87127 



87192 
87256 
87320 
87384 
87448 



87512 
87576 
87640 
87704 
87768 



87832 
87895 
87959 
88023 
88086 



88150 
88213 
88277 
88340 
88404 



88467 
88530 
88593 
88656 
88720 



88783 
88846 
88909 
88971 
89034 



S9097 
89160 
89223 
89285 
89348 



89411 
89473 
89536 
89598 
89660 



89723 
89785 
89847 
89910 
89972 



8-00034 
Lg. Vers. 



89506 
89576 
89646 
89716 
89786 



89856 
89926 
89995 
90065 
90135 



90205 
90274 
90344 
90413 
90483 



90552 
90822 
90691 
90760 
90830 



90968 
91037 
91106 
91175 



91244 
91313 
91382 
91451 
91520 



91588 
91657 
91728 
91794 
91863 



91932 
92000 
92068 
92137 
92205 




92955 
93022 
93090 
93158 
93226 




Log.Exs, 



69 



69 



8.90034 
90096 
90158 
90220 
90282 



90344 
90406 
90467 
90529 
90591 

90852 
90714 
90776 
90837 
90899 



90960 
91021 
91083 
91144 
91205 



91267 
91328 
91389 
91450 
91511 



91572 
91633 
91694 
91755 
91815 



91876 
91937 
91997 
9205& 
92119 



92179 
92240 
92300 
92361 
92421 



92487 
92542 
92602 
92662 
92722 



92782 
92842 
92902 
92962 
93022 




93381 
93440 
93500 
93560 
93619 



8-93679 
Lg. Vers, 



8.93631 
.93699 
.93766 
.93833 
.93901 



8.93968 
.94035 
•94102 
.94170 
.94237 



8.94304 
.94371 
.94438 
.94505 
.94572 



8.94638 
.94705 
.94772 
•94839 
•94905 



8.94972 
.95039 
•95105 
•95172 
•95238 



8.95305 
.95371 
.95437 
.95504 
.95570 



.95636 
.95703 
•95769 
•95835 
•95901 



.95967 
.96033 
.96099 
•96165 
.96231 




8.96953 
.97018 
.97084 
.97149 
.97214 



8.97280 
.97345 
.97410 
.97475 
.97540 



8-97608 
Log.Exs. 



P.P. 





70 


69 


6 


7.0 


6.9 


7 


8.1 


8.0 


8 


9.3 


9.2 


9 


10.5 


10.3 


10 


11.6 


11.5 


20 


23.3 


23.0 


30 


35-0 


34.5 


40 


46.6 


46.0 


50 


58.3 


57.5 





67 


66 


6^ 


6 


6.7 


6.6 


6. 


7 


7.8 


7.7 


7. 


8 


8.9 


8.8 


8. 


9 


10.0 


9.9 


9. 


10 


11.1 


11.0 


10. 


20 


22.3 


22.0 


21. 


30 


33.5 


33-0 


32. 


40 


44.6 


44-0 


43- 


50 


55.8 


55.0 


54. 



68 

6.8 

7-9 
9.0 
10.2 
11.3 
22.6 
34.0 
45.3 
56.6 





64 


63 


6 


6.4 


6-3 


7 


7 


4 


7.3 


8 


8 


5 


8.4 


9 


9 


6 


9.4 


10 


10 


6 


10.5 


20 


21 


3 


21.0 


30 


32 





31.5 


40 


42 


6 


42.0 


50 


53 


3 


52.5 





61 


60 


6 


6-1 


6-0 


7 


7.:. 


7-0 


8 


8. 


8-0 


9 


9.: 


9-0 


10 


10.: 


10-0 


20 


20-3 


20-0 


30 


30.5 


30.0 


40 


40.6 


40.0 


50 


50.8 


50.0 



63 

6.2 

7.2 

8.2 

9-3 

10-3 

20-6 

31-0 

41.3 

51.6 



59 

5-9 

6-9 

7-8 

8-8 

9.8 

19-6 

29-5 

3 

49.1 



6 

7 

8 

9 

10 

20 

30 

40 

50 



8^ 

o^o 

0.1 
O.T 
0.2 
0.3 
0.4 



P.P. 



661 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
34° 35° 



Lg. Vers. D Log.Exs. I> Lg. Vers, J> Log.Exs. I> 



8-93679 
93738 
93797 
93857 
93916 




93975 
94034 
94094 
94153 
94212 



94859 
94917 
94976 
95034 
95093 



95151 
95210 
95268 
95326 
95384 



95443 
95501 
95559 
95617 
95675 



95733 
95791 
95849 
95907 
95965 



96023 
96080 
96138 
96196 
96253 



96311 
96368 
96426 
96483 
96541 



96598 
96656 
96713 
96770 
96827 



96885 
96942 
96999 
97056 
97113 



8-97170 
Lg. Vers 



8.97606 
•97671 
.97736 
.97801 
.97865 



8.97930 
.97995 
.98060 
.98125 
.98190 



8.98254 
.98319 
.98383 
.98448 
.98513 



8.98577 
.98642 
.98706 
.98770 
.98835 



8.98899 
.98963 
.99028 
.99092 
.99156 



8.99220 
.99284 
.99348 
.99412 
•99476 



8.99540 
•99604 
.99668 
.99732 
.99796 



8.99860 

.99923 

8.99987 

9-00051 

•00114 



9.00178 
•00242 
•00305 
•00369 
•00432 




9^01128 
•01191 
•01254 
•01317 
•01380 



9-01443 
Log.Exs. 



97170 
97227 
97284 
97341 
97398 
97455 
97511 
97568 
97625 
97681 



97738 
97795 
97851 
97908 
97964 



98020 
98077 
98133 
98190 
98246 



98302 
98358 
98414 
98470 
98527 




99141 
99197 
99252 
99308 
99363 



99419 
99474 
99529 
99585 
99640 



99695 
99751 
99806 
99861 
99916 



99971 
00026 
00081 
00136 
00191 

00246 
00301 
00356 
00411 
00466 



00520 



Lg. Vers, 



9.01443 
•01505 
•01568 
•01631 
•01694 




•02382 
• 02444 
•02506 
•02569 
•02631 



•02693 
•02755 
.02817 
•02880 
•02942 



•03004 
•03066 
.03128 
•03190 
.03252 



9.03313 
•03375 
.03437 
.03499 
•03561 



9^03622 
•03684 
•03746 
•03807 
•03869 



9 •03930 
•03992 
•04053 
•04115 
•04176 



9 •04238 
•04299 
•04360 
•04421 
•04483 



9-04544 
•04605 
•04666 
•04727 
•04788 



9.04850 
•04911 
•04972 
•05033 
•05093 



9^05154 
Log.Exs. 



40 

41 
42 
43 
44 

45 
46 
47 
48 
-49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
00 



P.P. 





65 


64 


6 


6^5 


6-4 


7 


7^6 


7-4 


8 


8.6 


8.5 


9 


9.7 


9.6 


10 


10.8 


10^6 


20 


21.6 


21^3 


30 


32.5 


32^0 


40 


43.3 


42^6 


50 


54.1 


53^3 





63 


61 


6 


6 2 


6-11 


7 


7 


2 


7 




8 


8 


2 


8 




9 


9 


3 


9 




10 


10 


3 


10 




20 


20 


fi 


20 


3 


30 


31 


c 


30 


5 


4C 


4] 


3 


40 


6 


50 


51 


6 


50 


8 



10 
20 
30 
40 
50 



59 

5.8 

6 

7 

8 

9 
19 

29 _ 
39^3 
49^1 



58 

5 

6 

7 

8 

9 
19 
29 
38 
48 



56 


55 


5-6 


5^5 


6^5 


6.4 


7^4 


7^3 


8^4 


8-2 


9^3 


9^1 


18^6 


18^3 


28^0 


27.5 


37-3 


36.6 


46-6 


45.8 



63 

6.3 

7.3 

8.4 

9.4 

10.5 

21.0 

31.5 

42.0 

52.5 



60 

6.0 

7.0 

8.0 

9.0 

10.0 

20.0 

30.0 

40.0 

50.0 



57 

5.7 

6o6 

7.6 

8.5 

9.5 

19.0 

28.5 

38.0 

47.5 



54 

5.4 

6^3 

7.2 

8.1 

9.0 

18.0 

27.0 

36.0 

45.0 



6 

7 
8 

9 
10 
20 
30 
40 
50 



O 

0.0 
0.0 
0.0 
0.1 
0.1 

9-1 

0.2 
0.3 
0.4 



P. P. 



662 



TABLEVIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANXa 
26° 37° 



10 

11 
12 
13 
14 



15 
16 
17 
18 
19 

20 

21 
22 

2; 

24 



2. 
26 
27 

21 
2i 

30 

31 

32 
83 
34 



85 
36 
87 
38 
39 

40 

41 
42 
43 
ii 
45 
46 
47 
48 

50 

51 
52 
53 
54 



55 
56 
57 
58 
59 

60 



Lg.Vers, 



9.00520 
00575 
00630 
00684 
00739 



00794 
00848 
00903 
00957 
01011 



01066 
01120 
01174 
01229 
01283 



01337 
01391 
01445 
01499 
01554 



01608 
01662 
01715 
01769 
01823 



01877 
01931 
01985 
02038 
02092 



02146 
02199 
02253 
02307 
02360 



02414 
02467 
02521 
02574 
02627 




02947 
03000 
03053 
03106 
03159 



03212 
03265 
03318 
03371 
03423 



03476 
03529 
03582 
03634 
03687 



03740 
.g.Vers, 



JO Log.Exs, J> Lg.Vers. I> Log.Exs. I> 



•05154 
•05215 
.05276 
•05337 
•0 5398 
•05458 
•05519 
.05580 
.05640 
•05701 



h05762 
.05822 
.05883 
.05943 
.06004 



06064 
06124 
06185 
06245 
06305 



.06366 
.06426 
.06486 
.06546 
.06606 






.08456 
.08515 
.08574 
.08634 
.08693 



9-08752 
-D Log.Exs 



•03740 
•03792 
.03845 
•03898 
•03950 



9-04002 
.04055 
.04107 
.04160 
•04212 



9^04264 
.04317 
.04369 
.04421 
.04473 




9.05045 
.05097 
.05148 
.05200 
•05252 



9.05303 
.05355 
.05407 
•05458 
.05510 



9.05561 
.05613 
•05664 
.05715 
•05767 



9.05818 
.05869 
.05921 
•05972 
•06023 




9^06584 
•06635 
.06686 
.06736 
•06787 



9 06838 
Lg. Vers, 



08752 
08811 
08870 
08929 
08988 



09047 
09106 
09164 
09223 
09282 



09341 
09400 
09458 
09517 
09576 



09634 
09693 
09752 
09810 
09869 



09927 
09986 
10044 
10102 
10161 



10219 
10278 
10336 
10394 
10452 



10511 
10569 
10627 
10685 
10743 



10801 
10859 
10917 
10975 
11033 



11091 
11149 
11207 
11265 
11323 




11957 
12015 
12072 
12129 
12187 



9^12244 



Log.Exs, 



P.P. 





61 


60 


6 


6^1 


6.0 


7 


7 


1 


7.0 


8 


8 


1 


8.0 


9 


9 


1 


9.0 


10 


10 


1 


10.0 


20 


20 


3 


20.0 


30 


30 


5 


30.0 


40 


40 


g 


40^0 


50 


50 


8 


50.0 





58 


6 


5.8 


7 


6.7 


8 


7.7 


9 


8.7 


10 


9.6 


20 


19^3 


30 


29^0 


40 


38.6 


50 


48.3 





55 


54 


6 


5.5 


5.« 


7 


6.4 


6.a 


8 


7.3 




9 


8.2 


8.1 


10 


9.1 


C.9 


20 


18.3 


18.0 


30 


27.5 


27.0 


40 


36.6 


38.0 


50 


45.8 


43.0 




P.P. 



663 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
28° 39° 



Lg.Vers. I> Log.Exs, 1> Lg.Vers. I> 



06838 
06888 
06939 
06990 
07040 



07091 
07141 
07192 
07242 
07293 



07343 
97393 
07444 
07494 
07544 



07594 
07644 
07695 
07745 
07795 



07845 
07895 
07945 
07995 
08045 



08095 
08145 
08195 
08244 
08294 



08344 
08394 
08443 
08493 
08543 



08592 
08642 
08691 
08741 
08790 



08840 
08889 
08939 
08988 
09087 



09087 
09136 
09185 
09234 
09284 



09333 
09382 
09431 
09480 
09529 



09578 
09627 
09676 
09725 
09774 



90982? 
Lg.Vers. 



I) 



12244 
12302 
12359 
12416 
12474 



12531 
12588 
12645 
12703 
12760 



12817 
12874 
12931 
12988 
13045 



13102 
13159 
13216 
13273 
13330 



13387 
13444 
13500 
13557 
13614 



13671 
13727 
13784 
13841 
13897 



13954 
14011 
14067 
14124 
14180 



14237 
14293 
14350 
14406 
14462 



14519 
14575 
14631 
14688 
14744 



14800 
14856 
14913 
14969 
15025 



15081 
15137 
15193 
15249 
15305 




9-15641 
Log.Exs. 



09823 
09872 
09920 
09969 
10018 



10067 
10115 
10164 
10213 
10261 



10310 
10358 
10407 
10455 
10504 



10552 
10601 
10649 
10697 
10746 



10794 
10842 
10890 
10939 
10987 



11035 
11083 
11131 
11179 
11227 



11275 
11323 
11371 
11419 
11467 



11615 
11562 
11610 
11658 
11706 



11754 
11801 
11849 
11897 
11944 



11992 
12039 
12087 
12134 
12182 



12229 
12277 
12324 
12371 
12419 



12466 
12513 
12560 
12608 
12655 



12702 



Lg. Vers. 



.Exs. 2> 



9.15641 
15697 
15752 
15808 
15864 



15920 
15975 
16031 
16087 
16142 



16198 
16254 
16309 
16365 
16420 



16476 
16531 
16587 
16642 
16698 



16753 
16808 
16864 
16919 
16974 



17029 
17085 
17140 
17195 
17250 




17856 
17910 
17965 
18020 
18075 



18130 
18185 
18239 
18294 
18349 



18403 
18458 
18513 
18567 
18622 



18676 
18731 
18786 
18840 
18894 



9-18949 



Log.Exs, 



P.P. 





57 


57 


6 


5.7 


5-7 


7 


6.7 


6.6 


8 


7.6 


7.6 


9 


8-6 


8.5 


10 


9.6 


9.5 


20 


19.1 


19.0 


30 


28.7 


28.5 


40 


38.3 


38.0 


50 


47.9 


47.5 





56 


55 


6 


5.6 


5.5 


7 


6 


5 


6 


5 


8 


7 


4 


7 


4 


9 


8 


4 


8 


3 


10 


9 




9 


2 


20 


18 


6 


18 


5 


30 


28 





27 


7 


40 


37 




37 





50 


46 


6 


46 


2 



56 

5.6 

6.6 

7.5 

8.5 

9.4 

18.8 

28.2 

37.6 

47.1 

55 

5.5 

6.4 

7.3 

8.2 

9.1 

18.3 

27. D 

36.6 

45.8 





54 


6 


5.4 


7 


6-3 


8 


7.2 


9 


82 


10 


9-1 


20 


18.1 


30 


27.2 


40 


36-3 


50 


45.4 



54 

5.4 

6.3 

7.2 

8.1 

3.0 

18.0 

27.0 

36.0 

45.0 





51 


50 


6 


5.1 


5.0 


7 


5.9 


5.9 


8 


6.8 


6.7 


9 


7.6 


7.6 


10 


8.5 


8.4 


20 


17-0 


16.8 


30 


25-5 


25-2 


40 


34.0 


33.6 


50 


42.5 


42.1 





49 


49 


6 


4.9 


4.91 


7 


5.8 


5 


7 


8 


6.6 


6 


5 


9 


7.4 


7 


3 


10 


8.2 


8 


1 


20 


16-5 


16 


3 


30 


24.7 


24 


5 


40 


33.0 


32 


6 


50 


41.2 


40 


8 



50 

50 

5.8 

6.6 

7.5 

8.3 

16.6 

25.0 

33.3 

41.6 

48 

48 

5.6 

6.4 

7.3 

8.1 

16.1 

24.2 

32.3 

40.4 





48 


47 


47 


6 


4.8 


4.7 


4.7 


7 


5 


6 


5 


5 


5 


5 


8 


6 


4 


6 


3 


6 


2 


9 


7 


2 


7 


1 


7 





10 


8 





7 


9 


7 


8 


20 


16 





15 


8 


15 


6 


80 


24 





23 


7 


23 


5 


40 


32 





31 


6 


31 


3 


P50 


40 





39 


6 


39 


1 



P.P. 



664 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
30° 31° 



Lg. Vers. I> Log.Exs. -Z> Lg. Vers. i> Log.Exs 



12702 
12749 
12796 
12843 
12890 

12937 
12984 
13031 
13078 
13125 



13172 
13219 
13266 
13313 
13359 



13406 
13453 
13500 
13546 
13593 



13639 
13686 
13733 
13779 
13826 
13872 
13919 
13965 
14011 
14058 



14104 
14151 
14197 
14243 
14289 



14336 
14382 
14428 

14^:474 

14:520 



14566 
14612 
14658 
14704 
14750 



14796 
14842 
14888 
14934 
14980 
15026 
15071 
15117 
15163 
15209 



15254 
15300 
15346 
15391 
15437 



9-15483 
Lg. Vers 




18949 
19003 
19058 
19112 
19167 



19221 
19275 
19329 
19384 
19438 



20033 
20087 
20141 
20195 
20249 



20303 
20357 
20411 
20465 
20518 



20572 
20626 
20680 
20733 
20787 



20841 
20894 
20948 
21002 
21055 



21109 
21162 
21216 
21269 
21323 



21376 
21430 
21483 
21537 
21590 



21643 
21697 
21750 
21803 
21857 



21910 
21963 
22016 
22070 
22123 



9-22176 
Log.Exs, 



15483 
15528 
15574 
15619 
15665 



15710 
15755 
15801 
15846 
15891 



15937 
15982 
16027 
16073 
16118 



16163 
16208 
16253 
16298 
16343 




17061 
17106 
17151 
17195 
1724P 



17284 
17329 
17373 
17418 
17462 



17507 
17551 
17596 
17640 
17684 



17729 
17773 
17817 
17861 
17906 



17950 
17994 
18038 
18082 
18126 



9 •18170 
Lg. Vers 



22176 
22229 
22282 
22335 
22388 



22441 
22494 
22547 
22600 
22653 



22706 
22759 
22812 
22865 
22918 



22971 
23024 
23076 
23129 
23182 



23235 
23287 
23340 
23393 
23446 



23498 
23551 
23603 
23656 
23709 



23761 
23814 
23866 
23919 
23971 



24024 
24076 
24128 
24181 
24233 



24285 
24138 
24390 
24442 
24495 



24547 
24599 
24651 
24704 
24756 



24808 
24860 
24912 
24964 
25016 



25068 
25120 
25172 
25224 
25276 



9-25328 
Log.Exs. 



D 



P.P. 





5? 


54 


6 


5.4 


5.4 


7 


6 


3 


6.3 


8 


7 


2 


7.2 


9 


8 


2 


8.1 


10 


9 




9.0 


20 


18 


\ 


18.0 


30 


27 


2 


27.0 


40 


36 


3 


36.0 


50 


45 


4 


45.0 





53 


52 


5$ 


6 


5.3 


5.2 


5. 


7 


6.2 


6.1 


6. 


8 


7.0 


7.0 


6. 


9 


7.9 


7.9 


7. 


10 


8.8 


8.7 


8. 


20 


17.6 


17.5 


17. 


30 


26.5 


26.2 


26. 


40 


35.3 


35-0 


34. 


50 


44.1 


43.7 


43. 





47 


47 


6 


4.7 


4.7 


7 


5.5 


5.5 


8 


6.3 


6.2 


9 


7.1 


7.0 


10 


7.9 


7.8 


20 


15.8 


15.6 


30 


23.7 


23-5 


40 


31.6 


31-3 


50 


39.6 


39.1 





46 


45 


6 


4-6 


4-51 


7 


5-3 


5 


3 


8 


6-1 


6 





9 


6-9 


6 


8 


10 


7-6 


7 


6 


20 


15-3 


15 


1 


30 


23-0 


22 




40 


30-6 


30 


3 


50 


38.3 


37 


9 



53 

5.3 

6.2 

7.1 

8.0 

8.9 

17.8 

26.7 

35-6 

44.6 



46 

4.6 

5.4 

6.2 

7.0 

7.7 

15-5 

23-2 

31.0 

38.7 



45 

4.5 

5.2 

6.0 

6.7 

7.5 

15.0 

22.5 

30.0 

37.5 



7 
8 
9 
10 
20 
30 
40 
50 



44 


4 


4 


5 


2 


5 


9 


6 


7 


7 


4 


14 


8 


22 


2 


29 


6 


37 


1 



44 

I:! 

5.8 
6.6 
7.3 
14.6 
22.0 
29.3 
36.6 



P.P. 



665 



VABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
33° 33° 



Lg. Vers. J> Log.Exs. -D Lg. Vers. I> Log.Exs. I> 



18170 
18214 
18258 
18302 
18346 



18390 
18434 
18478 
18522 
18566 



18610 
18654 
18697 
18741 
18785 



18829 
18872 
18916 
18959 
19003 



19047 
19090 
19134 
19177 
19221 



19264 
19308 
19351 
19395 
19438 



19481 
19525 
19568 
19611 
19654 



19698 
19741 
19784 
19827 
19870 



19914 
19957 
20000 
20043 
20086 



20129 
20172 
20215 
20258 
20301 



20343 
20386 
20429 
20472 
20515 



20558 
20600 
20643 
20686 
20728 
9-20771 
Lg. Vers 



9.25328 
.25380 
.25432 
.25484 
■25536 



9.25588 
.25640 
.25692 
.25743 
.25795 



9.25847 
.25899 
.25950 
.26002 
•26054 



9-26105 
.26157 
.26209 
.26260 
.26312 



9.26364 
.26415 
.26467 
.26518 
.26570 



9.26621 
.26673 
.26724 
.26776 
■26827 



9.26878 
.26930 
.26981 
.27032 
.27084 



9.27135 
•27186 
.27238 
.27289 
■27340 



9.27391 
.27443 
.27494 
■27545 
■27596 



9.27647 
.27698 
.27749 
.27800 
.27852 



9.27903 
.27954 
.28005 
.28056 
.28107 



9.28157 
.28208 
.28259 
•28310 
•28361 
9-28412 
Log.Exs, 



9.20771 
20814 
20856 
20899 
20942 



20984 
21027 
21069 
21112 
21154 



21196 
21239 
21281 
21324 
21366 



21408 
21451 
21493 
21535 
21577 



21620 
21662 
21704 
21746 
21788 



21830 
21872 
21914 
21956 
21998 



22040 
22082 
22124 
22166 
22208 



22250 
22292 
22334 
22376 
22417 



22459 
22501 
22543 
22584 
22626 



22668 
22709 
22751 
22792 
22834 




Lg. Vers 



28412 
28463 
28514 
28564 
28615 



28666 
28717 
28768 
28818 
28869 



28920 
28970 
29021 
29072 
29122 



29173 
29223 
29274 
29324 
29375 



29426 
29476 
29527 
29577 
29627 



29678 
29728 
29779 
29829 
29879 



42 
42 
42 
43 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
4l 
42 
42 
4l 

42 
41 
42 
4l 
41 

42 
4l 
4l 
4l 
4l 
41 
41 
41 
41 
41 
41 
41 
41 
4l 
41 
41 

l>lLog.Exs 



29930 
29980 
30030 
30081 
30131 



30181 
30231 
30282 
30332 
30382 



30432 
30482 
30533 
30583 
30633 



30683 
30733 
30783 
30833 
30883 



30933 
30983 
31033 
31083 
31133 






1 

2 

3 

j4 

5 

6 

7 

8 

_9_ 

10 

11 

12 

13 

14 

15 

16 

17 

18 

H 

30 

21 

22 

23 

24 

25 
26 
27 
28 
29 

30 
31 
32 
33 
3i 
35 
36 
37 
38 
89 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
38 
59. 

60 



P. P. 





53 


5T 


6 


5.2 


5 11 


7 


6 





6 





8 


6 


9 


6 


3 


9 


7 


8 


7 


7 


10 


8 




8 


6 


20 


17 


3 


17 


1 


30 


26 





25 


7 


40 


34 


6 


34 


3 


50 


43 


3 


42 


9 





50 


50 


6 


5.0 


5-01 


7 


5 


9 


5 


8 


8 


6 


7 


6 


6 


9 


7 


6 


7 


5 


10 


8 


4 


8 


3 


20 


16 


8 


16 


g 


30 


25 


2 


25 





40 


33 


6 


33 


3 


50 


42 


1 


41 


6 





44 


43 


6 


4.4 


4.3 


7 


5 


1 


5 


1 


8 


5 


8 


5 


8 


9 


6 


6 


6 


5 


10 


7 


3 


7 


2 


20 


14 


6 


14 


5 


30 


22 





21 


7 


40 


29 


3 


29 





50 


36 


6 


36 


2 



43 


^:3 


4.2 


4.2 


4.9 


4.9 


5.6 


5.6 


6.4 


6.3 


7.1 


7.0 


14.1 


14.0 


21.2 


21.0 


28-3 


28.0 


35^4 


35.0 



51 

5.1 

5.9 

6.8 

7.6 

8.5 

17.0 

25.5 

34.0 

42.5 



49 

4.9 

5.8 

6.6 

7-4 

8^2 

16^5 

24-7 

33^0 

41.2 



43 

4.3 

5.0 

5-7 

6-4 

7.1 

14.3 

21.5 

28 ■ 6 

35.8 



41 

41 

4-8 

5.5 

6.2 

6 9 

13. i^ 

20^7 

27.6 

34.6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



41 

4.1 

48 

5.4 

6.1 

6.8 

13.6 

20.5 

27.3 

34.1 



P. P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
34° 35° 



Lg. Vers, 



23290 
23331 
23372 
23414 
23455 



23496 
23537 
23579 
23620 
23661 



23702 
23743 
23784 
23825 
23866 



23907 
23948 
23989 
24030 
24071 



24112 
24153 
24194 
24235 
24275 



24316 
24357 
24398 
24438 
24479 



24520 
24561 
24601 
24642 
24682 



24723 
24764 
24804 
24845 
24.885 



24926 
24966 
25007 
25047 
25087 



25128 
25168 
25209 
25249 
25289 



25329 
25370 
25410 
25450 
25490 



25531 
25571 
25611 
25651 
25691 
25731 
y, Vers. 



Log.Exs. 



.31432 
•31482 
.31532 
•31582 
■31632 



•31681 
•31731 
•31781 
.31831 
.31880 



•31930 
.31980 
.32029 
•32079 
■32129 

■32178 
•32228 
•32277 
•32327 
■32377 



9 •32426 
.32476 
.32525 
.32575 
•32624 




9.33167 
•33216 
•33266 
•33315 
■33364 



9 •33413 
.33463 
•33512 
•33561 
•33610 



9 •33659 
.33708 
.33758 
.33807 
.33856 



9.33905 
.33954 
.34003 
.34052 
.34101 



9.34150 
.34199 
.34248 
•34297 
•34346 



9-34395 
Log.Exs. 



.Vers. I> Log.Exs, 2> 



•25731 
•25771 
•25811 
•25851 
■25891 



•25931 
•25971 
•26011 
•26051 
•26091 



■26131 
•26171 
.26210 
.26250 
■26290 




•26726 
•26766 
•26806 
•26845 
■26885 



•26924 
•26964 
•27003 
.27042 
■27082 



•27121 
•27161 
•27200 
•27239 
■27278 




9.27709 
•27749 
•27788 
•27827 
•27866 



9^27905 
•27944 
•27982 
•28021 
•28060 



9-28099 
Lg. Vers. 



9 ■34639 
•34688 
•34737 
•34785 
•34834 



9-34395 
• 34444 
•34492 
•34541 
■34590 



9-34883 
.34932 
.34980 
•34029 
•35078 



■35127 
•35175 
•35224 
•35273 
■35321 





9-36581 
.36629 
.36678 
.36726 
■36774 



9.36822 
•36870 
.36919 
.36967 
•37015 



9 •37063 
•37111 
•37159 
•37207 
•37255 



9-37303 
Log.Exs. 



P.P. 





50 


49 


6 


5^0 


4-91 


7 


5-8 


5 


8 


8 


6-6 


6 


6 


9 


7-5 


7 


4 


10 


8-3 


8 


2 


20 


16-6 


16 


5 


30 


25-0 


24 


7 


40 


33-3 


33 





50 


41-6 


41 


2 



16-1 
24.2 
32.3 
40 .4] 



49 

4.9 

5^7 

6-5 

7^3 

8.1 

16^3 

24^5 

32^6 

40^8 

48 

4^8 

5. 





41 


41 


6 


4^1 


4-1 


7 


4^8 


4 


8 


8 


5^5 


5 


4 


9 


6-2 


6 


1 


10 


6.9 


6 


8 


20 


13-8 


13 


5 


30 


20^7 


20 


5 


40 


27-6 


27 


3 


50 


34-6 


34 


1 





40 


6 


4-0 


7 


4-7 


8 


5^4 


9 


6.1 


10 


6.7 


20 


13.5 


30 


20.2 


40 


27.0 


50 


33-7 



6 
7 
8 
9 

10 
20 
30 
40 
50 



40 

^.0 

4.; 

5.g 

6^0 

6.6 

13^3 

20^0 

26^6 

33^3 



39_ 39 

9 
5 
2 
8 
5 

5 




3^9 


3. 


4 


6 


4- 


5 


2 


5- 


5 


9 


5- 


6 


g 


6. 


13 


1 


13 ■ 


19 


7 


19^ 


26 


3 


26 ■ 


32 


9 


32. 



6 
12 
19 
25 
32 
P.P. 



667 



'^ABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
36° 37° 



Lg. Vers. I> Log.Exs. I> Lg. Vers. I> Log.Exs. I> 



o 


9.28099 


1 


.28138 


2 


.28177 


3 


.28816 


4 


.28255 


5 


9.28293 


6 


.28332 


7 


•28371 


8 


.28410 


9 


.28448 


10 


9.28487 


11 


.28526 


12 


.28564 


13 


.28603 


14 


.28642 


15 


9.28680 


16 


.28719 


17 


.28757 


18 


.28796 


,19 


.28835 


20 


9.28873 


21 


.28912 


22 


.28950 


23 


.28988 


24 


.29027 



25 
26 
27 
28 
29 




P.P. 





48 


6 


4.8 


7 


5.6 


8 


6.4 


9 


7.3 


10 


8.1 


20 


16.1 


30 


24.2 


40 


32.3 


50 


40.4 





47 


6 


4-7 


7 


55 


8 


6.3 


9 


7.1 


10 


7.9 


20 


15.8 


30 


23.7 


40 


31.6 


50 


39.6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



48 

48 

5.6 

6.4 

7.2 

8.0 

16. 

24.0 

32.0 

40.0 

47 

4.7 

5.5 

6.2 

7.0 

7.8 

15-6 

23.5 

31.3 

39.1 

46. 

4.6 

5.4 

6.2 

70 

7.7 
15.5 
23.2 
31.0 
38.^ 



6 
7 
8 

9 
10 

20 
30 
40 
50 



39 

39 

4-5 

5.2 

5-8 

6.5 

13.0 

10.5 

26.0 

32.5 





38 


6 


8.8 


7 


4.4 


8 


5.0 


9 


5.7 


10 


6.3 


20 


12.6 


30 


19.0 


40 


25.3 


50 


31.6 





37 


6 


3.7 


7 


4.3 


8 


4.9 


9 


5.5 


10 


6.1 


20 


12.3 


80 


18-5 


40 


24.6 


FO 


30. P 



38. 

3.8 

4.5 

5.1 

5.8 

6.4 

12.8 

19.2 

25.6 

32.1 

37_ 

3.7 

4.4 

5.0 

5-6 

6.2 

12.5 

18.7 

25.0 

31.2 

36„ 

3.6 
4-2 
4.8 

5.5 
6.1 
12.1 
18.2 
24.3 
30.4 



P.P. 



668 



TABLE VIII.—LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
38° 39° 



Lg. Vers, 



32631 
32668 
32704 
32741 
32778 



32814 
32851 
32888 
32924 
32961 

32997 
33034 
33070 
33107 
33143 



33180 
33216 
33252 
33289 
33325 



33361 
33398 
33434 
33470 
33507 



33543 
33579 
33615 
33652 
33688 



33724 
33760 
33796 
33833 
33869 



33905 
33941 
33977 
34013 
34049 



34085 
34121 
34157 
34193 
34229 



34265 
34301 
34337 
34373 
34408 



34444 
34480 
34516 
34552 
34587 



34623 
34659 
34695 
34730 
34766 



9-34802 
Lg. Vera 



D 



Log.Exs. 



9-42978 
.43024 
.43071 
.43118 
.43164 



9.43211 
.43257 
.43304 
.43350 
.43396 



9.43443 
.43489 
.43536 
.43582 
.43629 



9.43675 
.43721 
.43768 
•43814 
•43861 



9-43907 
.43953 
.43999 
.44046 
.44092 



9.44138 
.44185 
.44231 
.44277 
.44323 



9.44370 
.44416 
.44462 
.44508 
.44554 



9.44601 
. 44847 
.44693 
.44739 
.44785 



9.44831 
•44877 
.44924 
•44970 
•45016 




Log.Exs. 



D 



Lg. Vers, 




9^35158 
•35193 
.35229 
.35264 
.35300 



9.35335 
.35370 
.35406 
.35441 
■35477 



9.35512 
.35547 
.35583 
.35618 
•35653 



9-35689 

.35724 
.35759 
.35794 
.35829 



9-35865 
.35900 
.3593^ 
.35970 
.36005 



9.36040 
.36076 
.36111 
.36146 
.36181 



9.36216 
.36251 
.36286 
.36321 
.36356 



•36391 
•36426 
•36461 
•36495 
•36530 



•36565 
•36600 
•36635 
•36670 
■36705 



9-36739 
•36774 
•36809 
•36844 
•36878 



9.3R913 
Lg. Vers. 



Log.Exs. 



45752 
45797 
45843 
45889 
45935 



45981 
46027 
46073 
46118 
46164 



46210 
46256 
46302 
46347 
46393 



46439 
46485 
46530 
46576 
46622 



46668 
46713 
46759 
46805 
46850 



46896 
46942 
46987 
47033 
47078 



47124 
47170 
47215 
47261 
47306 



47352 
47398 
47443 
47489 
47534 



47580 
47625 
47671 
47716 
47762 



47807 
47852 
47898 
47943 
47989 



48034 
48080 
48125 
48170 
48216 



48261 
48306 
48352 
48397 
48442 



9-48488 



Log.Exs. 



X) 



P.P. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



47 



4 


7 


5 


5 


6 


2 


7 





7 


8 


15 


6 


23 


5 


31 


3 


39 


1 





46, 


6 


4-61 


7 


5 


3 


8 


6 


1 


9 


6 


9 


10 


7 


6 


20 


15 


3 


30 


23 





40 


30 


6 


50 


38 


3 



46_ 

4^6 

5-4 

6-2 

7.0 

7.7 

15-5 

23.2 

31.0 

38.7 

45 

4.5 

5.3 

6 

6 8 

7.6 

15.1 

22.7 

30.3 

37.9 



45 

4.5 

5.2 

6.0 

6.7 

7-5 

15-0 

22-5 

30.0 

37.5 





37 


6 


3-71 


7 


4 


3 


8 


4 


9 


9 


5 


5 


10 


6 


1 


20 


12 


3 


30 


18 


5 


40 


24 


6 


50 


30 


8 



e 

7 

p 

9 
10 
20 
30 
40 
50 



36 

3^6 

4^2 

4^8 

5.4 

6^0 

12^0 

18^0 

24^0 

30. Oj 



35 


3 


5 


4 


1 


4 


6 


5 
5 


2 
8 


11 


6 


17 


5 


23 


3 


29 


ll 



36 

3.6 

4.2 

4.8 

5.5 

6.1 

12.1 

18.2 

24.3 

30.4 

35 

3.5 

41 

4.7 

5.3 

5.9 

11.8 

17.7 

23.6 

29.6 

3l 

3.4 

4.0 

4.6 

5.2 

5.7 

11^5 

17^2 

23-0 

98.7 



P.P. 



669 



TABLE VIII.— LOGARITHMIC VERSED SINES ANt) EXTERNAL SECANTS. 
40^ 41° 



Lg. Vers. 




36913 
36948 
36982 
37017 
37052 



37086 
37121 
37156 
37190 
37225 



37259 
37294 
37328 
37363 
37397 



37432 
37466 
37501 
37535 
37570 



37947 
37982 
38016 
38050 
38084 



38118 
38153 
38187 
38221 
38255 



38289 
38323 
38357 
38391 
38425 



38459 
38493 
38527 
38561 
38595 



38629 
38663 
38697 
38731 
38765 



38799 
38833 
38866 
38900 
38934 
389BR 



Lg. Vers 



JD 



9.491 _ 
.49211 
.49257 
.49302 
.49347 



J> 



Log.Exs, 



9.48488 
.48533 
•48578 
.48624 
.48669 




9.49392 
.49437 
.49482 
.49527 
.49572 



9.49618 
•49663 
.49708 
.49753 
.49798 



9.49843 
.49888 
•49933 
•49978 
.50023 



9.50068 
.50113 
.50158 
.50203 
.50248 



9.50293 
•50338 
•50383 
•50427 
.50472 



9.50517 
•50562 
•50607 
•50652 
.50697 



9.50742 
.50787 
.50831 
.50876 
•50921 



9.50966 
.51011 
.51055 
.51100 
•51145 

9.51190 



Log.Exs 



Lg. Vers, 



38968 
39002 
39035 
39069 
39103 



39137 
39170 
39204 
39238 
39271 



39305 
39339 
39372 
39406 



39473 
39507 
39540 
39574 
39607 



39641 
39674 
39708 
39741 
39774 



39808 
39841 
39875 
39908 
39941 



39975 
40008 
40041 
40075 
40108 



40141 
40175 
40208 
40241 
40274 



40307 
40341 
40374 
40407 
40440 



40473 
40506 
40540 
40573 
40606 



40639 
40672 
40705 
40738 
40771 



40804 
40837 
40870 
40903 
40936 
40969 



jy Lg.Vers, 



D 



Log.Exs, 



.51190 
.51235 
.51279 
.51324 
•51369 



•51414 
.51458 
.51503 
.51548 
•51592 



•51637 
.51682 
.51726 
.51771 
•51816 



•51860 
•51905 
•51950 
•51994 
.52039 



•52084 
•52128 
•52173 
•52217 
•52262 
•52306 
•52351 
•52396 
• 52440 
•52485 



•52529 
.52574 
•52618 
.52663 
•52707 





•53640 
•53684 
.53728 
•53773 
•53817 



9-53861 



I> iLog.Exs 
670 



2) 



n 



29 
30 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
ii 
45 
46 
47 
48 
49 
50 
51 
52 
53 
M 
55 
56 
57 
58 
A9 
60 



P.P. 





45 


6 


4^5 


7 


5-3 


8 


6.0 


9 


6.8 


10 


76 


20 


15-1 


30 


22.7 


40 


30^3 


50 


37.9 





41 


6 


4.4 


7 


5.2 


8 


5-9 


9 


6.7 


10 


7.4 


20 


14-8 


30 


22.2 


40 


29-6 


50 


37.1 





35 


6 


3.5 


7 


4.1 


8 


4.6 


9 


5.2 


10 


5-8 


20 


11.6 


30 


17-5 


40 


23.3 


50 


29.1 





34 


6 


3.4 


7 


3-9 


8 


4-5 


9 


5^1 


10 


5-6 


20 


11-3 


30 


17.0 


40 


22.6 


50 


28.3 



6 
7 
8 

9 
10 
20 
30 
40 
50 



45 

4.5 
5-2 
60 

6.7 
7.5 

15.0 
22.5 
30.0 
37-5 



44 

4.4 

5.1 

5-8 

6.6 

7.3 

14.6 

22^0 

29. S 

36.6 



31. 

3.4 

4.0 

4.6 

5.2 

5.7 

11.5 

17^2 

23^0 

28.7 



33 

3.3 

3.9 

4.4 

5.0 

5^6 

ll-I 

16. Z 

22.3 

27.9 



33 

3.3 
3.8 

4^4 
4.9 
5.5 
11.0 
16.5 
22.0 
27.5 



P.P. 



fABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
43° 43° 



9-40969 
41001 
41034 
41067 
41100 



Lg. Vers. 



41133 
41166 
41199 
41231 
41264 

41297 
41330 
41382 
41395 
41428 



41461 
41493 
41526 
41559 
41591 



41624 
41657 
41689 
41722 
41754 



41787 
41819 
41852 
41885 
41917 



41950 
41982 
42014 
42047 
42079 



42112 
42144 
42177 
42209 
42241 



42274 
42306 
42338 
42371 
42403 



42435 
42467 
42500 
42532 
42564 
42596 
42629 
42661 
42693 
42725 



42757 
42789 
42822 
42854 
42886 



9.42918 
Lg. Vers. 



Log.Exs, 



9.53861 
53906 
53950 
53994 
54038 



54083 
54127 
54171 
54215 
54259 



54304 
54348 
54392 
54436 
54480 



54525 
54569 
54813 
54657 
54701 



54745 
54790 
54834 
54878 
54922 



54966 
55010 
55054 
55098 
55142 



55186 
55230 
55275 
55319 
55363 



55407 
55451 
55495 
55539 
55583 



55627 
55671 
55715 
55759 
55803 



55847 
55890 
55934 
55978 
56022 



56066 
56110 
56154 
56198 
56242 



56286 
56330 
56374 
56417 
56461 



9.56505 
Log.Exs, 



1> 



Lg. Vers, 



42918 
42950 
42982 
43014 
43046 



43078 
43110 
43142 
43174 
43206 



43238 
43270 
43302 
43334 
43365 



43397 
43429 
43461 
43493 
43525 



43557 
43588 
43620 
43652 
43684 



43715 
43747 
43779 
43810 
43842 



43874 
43906 
43937 
43969 
44000 



44032 
44064 
44095 
44127 
44158 



44190 
44221 
44253 
44284 
44316 



44347 
44379 
44410 
44442 
44473 



44504 
44536 
44567 
44599 
44630 



44661 
44693 
44724 
44755 
44787 



9.44818 
Lg. Vers 



D 



32 
32 
32 
32 

32 
32 
31 
32 
32 
32 
32 
32 
32 
31 
32 
32 
32 
31 
32 
32 
31 
32 
31 
32 

31 
32 
31 
31 
32 

31 
32 
31 
UI 
31 

32 
31 
31 
31 
31 

31 
31 
31 
31 
31 

31 
31 
31 
31 
31 



Log.Exs, 




56505 
56549 
56593 
56637 
56680 



57162 
57206 
57250 
57293 
57337 



57381 

57424 
57468 
57512 
57556 



57599 
57643 
57687 
57730 
57774 



57818 
57861 
57905 
B7949 
57992 



58036 
58079 
58123 
58167 
58210 



58254 
58297 
58341 
58385 
58428 




58907 
58951 
58994 
59037 
59081 



9-59124 



Log.Exs. 



D 



P.P. 



10 
20 
30 
40 
50 



10 
20 
30 
40 
50 



7 
8 
9 
10 
20 
30 
40 
50 



43_ 



4 


4 


4. 


5 


2 


5. 


5 


9 


5. 


6 


7 


6. 


7 


4 


7. 


14 


8 


14. 


22 


2 


22. 


29 


6 


29. 


37 


1 


36. 



43. 

4.3 



44 

4 
1 
8 
6 
3 
6 

3 
6 



43 

4. 





33 


6 


3.3 


7 


3 


8 


8 


4 


4 


9 


4 


9 


10 


5 


5 


20 


11 





30 


16 


5 


40 


22 





50 


27 


5 



33 



3 


2 


3 


7 


4 


2 


4 


8 


5 


3 


10 


6 


16 





21 


3 


26 


6 



33 

3.2 
3-8 

4.3 
4.9 
5.4 
10.8 
16-2 
21-6 
27.1 



31 

3-1 

3.7 

4.2 

4.7 

5.2 

10.5 

15.7 

21.0 

26-2 



6 


3. 


7 


3. 


8 


4. 


9 


4- 


10 


5. 


20 


10. 


30 


15. 


40 


20. 


50 


25. 



31 

1 
6 
1 
6 
1 
3 
5 
6 
8 



P. P. 



671 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS J 
44° 45° 



Lg.Vers. 



O 

1 

2 

3 

_4 

5 
6 
7 
8 
_9 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 



9. 44818 
•44849 
.44880 
.44912 
.44943 



20 

21 
22 
23 
24 



25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 



•44974 
.45005 
.45036 
.45068 
.45099 



.45130 
.45161 
.45192 
•45223 
.45254 



•45285 
.45316 
•45348 
•45379 
•45410 



•45441 
• 45472 
•45503 
•45534 
•45565 



•45595 
•45626 
•45657 
•45688 
•45719 



•45750 
•45781 
•45812 
•45843 
•45873 



40 

41 
42 
43 
41 
45 
46 
47 
48 
49 



9-46058 
•46089 
•46120 
•46150 
•46181 



•45904 
•45935 
•45966 
•45997 
•46027 



2> 



9 •46212 
•46242 
•46273 
.46304 
•46334 



9-46365 
•46396 
•46426 
•46457 
•46487 



50 

51 
52 
53 
54 

55 9-46518 

56 -46549 



60 



-46579 
-46610 
•46640 



9-46671 
Lg. Vers 



31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
31 
3l 
31 
31 
31 
31 
31 
31 
31 
30 
31 
31 
31 
31 

31 
30 
31 
31 
30 

31 
31 
30 
31 
30 

31 
30 
31 
30 
31 
30 
30 
31 
30 
30 

31 
30 
30 
30 
30 

31 
30 
30 
30 
30 
30 



Log.Exs, 





9-59993 
.60036 
.60079 
.60123 
.60166 



9.60209 
•60253 
.60296 
.60339 
.60383 



9 . 60426 
.60469 
.60512 
.60556 
•60599 



2> 



Lg. Vers. 



9^60642 
60685 
60729 
60772 
60815 



9.60858 
.60902 
.60945 
.60988 
•61031 



9 •61075 
•61118 
.61161 
.61204 
•61247 



9^61291 
.61334 
•61377 
•61420 
•61463 



9^61506 
•61550 
•61593 
•61636 
-61679 



9-61722 
Log.Exs. 



-46671 
.46701 
.46732 
.46762 
.46793 



9-46823 
.46853 
.46884 
.46914 
•46945 



9^46975 
.47005 
.47036 
.47066 
-47096 



9-47127 
•47157 
.47187 
.47218 
-47248 



.47278 
.47308 
•47339 
•47369 
.47399 



9-47429 
.47459 
.47490 
•47520 
.47550 



2> 



Log.Exs, 



9.47580 
•47610 
•47640 
.47670 
•47700 



9^47731 
•47761 
•47791 
•47821 
•47851 



9 •47881 
.47911 
.47941 
.47971 
•48001 



•48031 
.48061 
.48090 
.48120 
•48150 



9 •48180 
•48210 
•48240 
•48270 
-48300 



I •48329 
-48359 
•48389 
•48419 
-48449 



9 48478 



i) Lg. Vers 



-61722 
.61765 
.61808 
.61852 
.61895 



-61938 
.61981 
.62024 
•62067 
-62110 



•62153 
•62196 
•62239 
•62282 
•62326 



•62369 
•62412 
•62455 
•62498 
•62541 



.62584 
.62627 
.62670 
.62713 
•62756 



2> 



•62799 
•62842 
•62885 
•62928 
•62971 



•63014 
•63057 
•63100 
•63143 
•63186 



•63229 
•63272 
•63315 
•63358 
•63401 



- 63443 
•63486 
•63529 
•63572 
•63615 



•63658 
•63701 
•63744 
•63787 
•63830 



•63873 
•63915 
•63958 
•64001 
• 64044 




i> Log.Exs 

672 



o 

1 

2 
3 

5 
6 
7 
8 
_9^ 

10 

11 
12 
13 
14 

15 
16 
17 
18 
11 
30 
21 
22 
23 
2£ 

25 
26 
27 
28 
29_ 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39, 

40 

41 
42 
43 
44 
45 
46 
47 
48 
ii 
50 
51 
52 
53 
54 
55 
56 
57 
58 
19 
60 



P.P. 





43 


6 


4^3 


7 


5^1 


8 


5.8 


9 


6.5 


10 


7.2 


20 


14^5 


30 


21^7 


40 


29^0 


50 


36^2 



43 

4.3 

5^0 

5^7 

6^4 

7-1 

14-3 

21^5 

28-6 

35^8 



42 

4-2 

4-9 

5^6 

6^4 

71 

14.1 

21.2 

28-3 

35^4 



31 

3^1 

• 7 

• 2 

• 7 

• 2 
•5 
•7 

• 

• 2 



31 

3^1 



30_ 



3^0| 


3 


5 


4 





4 


6 


5 




10 


1 


15 


2 


20 


3 


25 


4 



30 

30 

3^5 

4-0 

4-5 

5^0 

10-0 

15^0 

20^0 

25^0 



29 

2^9 
3^4 
3.9 
4.4 
4.9 



14^7 
19^6 
24-6 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
46° 47° 



Lg. Vers. 



9.48478 
•48508 
.48538 
.48568 
.48597 



J> 



9.48627 
.48657 
.48886 
.48716 
■48746 



9.48775 
.48805 
.48835 
.48864 
.48894 



9-48923 
.48953 
.48983 

.49012 
•49049 



9.49071 
.49101 
.49130 
.49160 
.49189 



•49219 
.49248 
.49278 
•49307 
•49336 



•49300 
•49395 
•49425 
•49454 
• 49483 
•49513 
•49542 
•49571 
•49601 
•49630 



•49059 
•49689 
•49718 
.49747 
•49776 



•49806 
•49835 
.49864 
.49893 
•49922 



•49952 
•49981 
•50010 
.50039 
.50068 



.50097 
.50126 
.50155 
.50185 
.50214 



9.50243 



' Lg. Vers, 



Log.Exs 



9.64301 

• 64344 
•64387 
. 64430 

• 64473 



9.64515 
.64558 
.64601 
. 64644 
.64687 



1.64729 
.64772 
•64815 
•64858 
.64901 



9.64943 
•64986 
.65029 
.65072 
.65114 



1.65157 
.65200 
.65243 
•65285 
.65328 



•65371 
•65414 
•65456 
•65499 
•65542 



•65585 
•65627 
•65870 
.65713 
•65755 



9.65798 
65841 
65884 
65926 
65969 



•66012 
•66054 
•66097 
.68140 
•66182 




•66651 
•66694 
•66737 
.66779 
_^668_22 
i.66RfiA 



Log, Exs, 



43 
42 
43 
43 
42 
43 
43 
42 
43 
42 
43 
43 
42 
43 
42 
43 
42 
43 
42 

43 
42 
43 
42 
43 
42 
43 
42 
43 
42 
43 
42 
43 
42 
42 

43 
42 
43 
42 
42 

43 
42 
42 
43 
42 

42 

43 
42 
42 
43 
42 
42 
42 
43 
42 

42 
42 
43 
42 
42 

42 



Lg. Vers 



9-50243 
.50272 
.50301 
•50330 
.50359 



9.50388 
•50417 
•50446 
•50475 
.50504 



9.50533 
.50562 
.50591 
.50619 
.50648 



9.51109 
■51138 
.51167 
.51195 
.51224 



9.51253 
.51281 
.51310 
.51338 
■51367 



9 ■50677 
.50706 
.50735 
.50764 
.50793 



9.50821 
.50850 
.50879 
.50908 
.50937 



9.50965 
•50994 
.51023 
•51052 
•51080 




■51681 
■5171C 
■51738 
■51767 
•51795 



9.51823 
•51852 
•51880 
•51909 
•51937 



9 .51965 
Lg. Vers 



29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
29 
28 
29 
29 
29 
28 
29 
29 
28 
29 
29 
28 
29 

28 
29 
28 
29 
28 
29 
28 
29 
28 
28 
29 
28 
28 
28 
29 

28 
28 
28 
28 
28 
29 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 



-og, Exs 



•66864 
•66907 
•66950 
•66992 
■67035 



1-67077 
■67120 
■67162 
■67205 
■67248 



■67290 
■67333 
-67375 
■67418 
■67460 



■67503 
•67546 
■67588 
•67631 
-67673 



2> 



■67716 
•67758 
•67801 
•67843 
■67886 



•67928 
•67971 
•68013 
•68056 
■68098 



■68141 
•68183 
•68226 
•68268 
•68311 



•68353 
-68396 
•68438 
•68481 
■68523 



■68566 
■68608 
•68651 
.68693 
■68735 



■68778 
■68820 
■68863 
■68905 
■68948 



■68990 
■69033 
■69075 
■69117 
■ 69160 
■69202 
■69245 
■69287 
■69330 
■69372 



9-69414 



Log.Exs, 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29. 
30 
31 
32 
33 
31 
35 
36 
37 
38 
3£ 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49, 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59_ 
fiO 



P. P. 



43 



10 
20 
30 
40 
50 



4 


3 


4- 


5 





4^ 


5 


7 


5^ 


6 


4 


6^ 


7 


1 


7. 


14 


3 


14 ■ 


21 


5 


21. 


28 


6 


28. 


35 


8 


35. 



43_ 

2 
9 
6 
4 
1 
1 
2 
3 
4 



6 


4. 


7 


4. 


8 


5. 


9 


6. 


10 


7. 


20 


14. 


30 


21. 


40 


28. 


50 


35. 



43 

2 
9 
6 
3 








30 



7 
8 
9 
10 
20 
30 
40 
50 



3 





3 


5 


4 





4 


5 


5 





10 





15 





20 





25 






39 

2.9 

3.4 

3-9 

4-4 

4^9 

9.8 

14-7 

19.6 

24.6 



39 



1 2 


9 


2 


3 


4 


3. 


3 


8 


3^ 


4 


3 


4. 


4 


3 


4. 


9 


6 


9. 


14 


5 


14. 


19 


3 


19. 


24 


1 


23. 



38 
8 
3 
8 
3 
7 
5 
2 

7 



10 
20 
30 
40 
50 



38 
2.8 

3.2 

3.7 

4.2 

4-6 

9.3 

14.0 

18.6 

23.3 



P. P. 



673 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
48° 49° 



10 

11 
12 
13 
li 
15 
16 
17 
18 
19 



9-52390 
.52418 
.52446 
.52474 
.52503 



30 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 



9.52531 
.52559 
.52587 
•52615 
.52643 



40 

41 
42 
43 
44 



45 
46 
47 
48 
49 
60 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Lg. Vers. J> 



9.51965 
• 51994 
. 52022 
.52050 
.52079 



9.52107 
•52135 
•52164 
•52192 
•52220 



9.52249 
•52277 
•52305 
•52333 
•52362 



9.52671 
.52699 
•52727 
•52756 
•52784 



9-52812 
•52840 
.52868 
.52896 
-52924 



9-52952 
52980 
53008 
53036 
53064 



9-53092 
•53120 
-53147 
.53175 
-53203 



-53231 
-53259 
-53287 
-53315 
-53343 



53370 
-53398 
-53426 
-53454 
-53482 



-53509 
-53537 
-53565 
-53593 
jJ362n 

•53648 



Log.Exs. D 



9.70262 
•70304 
•70347 
.70389 
•70431 



9 . 70474 
•70516 
.70558 
.70601 
♦70643 



' Lg.VersJJ^ 



9.69414 
•69457 
•69499 
•69542 
.69584 



9.69626 
•69669 
•69711 
•69753 
•69796 



9 •69838 
•69881 
•69923 
•69965 
-70008 



9-70050 
■70092 
•70135 
•70177 
-70220 



9.70685 
•70728 
•70770 
•70812 
•70854 



9-70897 
•70939 
•70981 
•71024 
-71066 



9.71108 
.71151 
.71193 
.71235 
.71278 



9.71320 
71362 
71404 
71447 
71489 



9.71531 
.71573 
•71616 
•71658 
•71700 



9-71743 
.71785 
-71827 
-71869 
-71912 



a 71954 
Log.Exs 



Lg. Vers, 



-53648 
.53676 
.53704 
•53731 
•53759 



.53787 
•53814 
•53842 
•53870 
•53897 



•53925 
•53952 
•53980 
•54008 
•54035 



• 54063 
•54090 
•54118 
•54145 
•54173 



. 54200 
•54228 
•54255 
•54283 
•54310 



.54338 
.54365 
.54393 
. 54420 
• 54448 



•54475 
•54502 
•54530 
•54557 
•54585 



.54612 
.54639 
.54667 
.54694 
■54721 



. 54748 
.54776 
. 54803 
.54830 
.54858 



.54885 
.54912 
.54939 
.54967 
-54994 



-55021 
.55048 
.55075 
-55103 
-55130 



9-55157 
-55184 
-55211 
•55238 
-55265 



9 55292 
Lg. Vers. 



D 



Log.Exs, 



71954 
71996 
72038 
72081 
72123 



72165 
72207 
72250 
72292 
72334 



72376 
72419 
72461 
72503 
72545 



72587 
72630 
72672 
72714 
72756 



72799 
72841 
72883 
72925 
72967 



73010 
73052 
73094 
73136 
73178 



73221 
73263 
73305 
73347 
73389 



73431 
73474 
73516 
73558 
73600 



73642 
73685 
73727 
73769 
73811 



73858 
73895 
73938 
73980 
74022 



74064 
74106 
74148 
74191 
74233 



74275 
74317 
74359 
74401 
74444 



7448R 
Log.Exs, 



Z> 



Z> 



P. P. 



43_ 43 

61 4-21 4-2 

7 4-9 4.9 

8 5.6 5.6 

9 6.4 6.3 
10 7-1 7.0 
20 14.1 14.0 
3021.2 21.0 
40 28.3 28.0 
50135. 4I35.O 



1 





28 


28 


6 


2-8 


2-8 


7 


3 


3 


3.2 


8 


3 


8 


3^7 


9 


4 


3 


4^2 


10 


4 


7 


4^6 


20 


9 


5 


9^3 


30 


14 


2 


14-0 


40 


19 





18-6 


50 


23 


7 


23^3 



27 

2-7 

3-2 

3-6 

4.1 

4.6 

9-1 

13-7 

18-3 

22.9 



27 

2^7 



P, P. 



674 



TABLE YHI.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTa 
50* 61* 



Lg.Vers, 



9.55292 
55319 
55347 
55374 
55401 



55428 
55455 
55482 
55509 
55536 



D 



55563 
55590 
55617 
55644 
55671 



55698 
55725 
55751 
55778 
55805 



55832 
55859 
55886 
55913 
55940 



55906 
55993 
56020 
56047 
56074 



56101 
56127 
56154 
56181 
56208 



56234 
56261 
56288 
56315 
56311 



56368 
56395 
56421 
56448 
56475 



56501 
56528 
56554 
56581 
56608 



56634 
56661 
56687 
56714 
56741 



58787 
56794 
56820 
56847 
56873 



9-56900 



Log.Exs, 



9.74486 
•74528 
. 74570 
.74612 
.74654 



9.74696 
.74739 
.74781 
.74823 
.74865 



I> 



9.74907 
.74949 
.74991 
.75033 
.75076 



9.75118 
.75160 
.75202 
.75244 
.7528 



9.75328 
.75370 
.75413 
.75455 
.75497 



.75539 
.75581 
.75623 
.75665 
.75707 



9.75750 
.75792 
.75834 
.75876 
.75918 



9.75960 
.76002 
.76044 
.76086 
.76128 



9.76171 
.76213 
.76255 
.76297 
.76339 



9.76381 
.76423 
.76465 
.76507 

.7654S 



9.78592 
.76634 
.76676 
.76718 
.76760 



9.76802 
.76344 
.768^6 
.76928 
.76970 



9-77012 
Log.Exs, 



42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 

42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 



Lg. Vers, 



9.56900 
.56926 
.56953 
.56979 
.57005 



9.57032 
.57058 
.57085 
.57111 
.57138 



9.57690 
.57716 
.57742 
.57768 
.57794 



9.57164 
.57190 
.57217 
.57243 
.57269 



9.57296 
.57322 
.57348 
.57375 
.57401 



2> 



9.57427 
.57454 
.57480 
.57506 
.57532 



9.57559 
.57585 
.57611 
.57637 
.57664 



9.57821 
.57847 
.57873 
.57899 
.57925 



9.57951 
57977 
58003 
58029 
58055 



9.58082 
58108 
58134 
58160 
58186 



9.58212 
.58238 
.58264 
.58290 
♦58316 



9.58342 
.58367 
.58393 
•58419 
■58445 



9-58471 



-2> Lg. Vers 



26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

26 
26 
26 
26 
26 

26 
26 
26 
26 
26 
26 
26 
26 
^6 
26 

26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
25 
26 
26 
26 
26 



Log.Exs 



9.77012 
.77055 
.77097 
•77139 
.77181 



9.77223 
•77265 
.77307 
.77349 
.77391 



9.77433 
.77475 
.77517 
.77560 
.77602 



9.78064 
.78107 
.78149 
.78191 
.78233 




J> 




9.78696 
.78738 
.78780 
.78822 
.78864 



9.78906 
.78948 
.78990 
.79032 
.79074 



9.79116 
.79158 
.79200 
.79242 
•79285 



9.79327 
.79369 
.79411 
•79453 
.79495 



9-79537 



Log.Exs, 



O 

1 
2 
3 
_4 
5 
6 
7 
8 
9 

10 

11 

12 
13 
14 

15 
16 
17 
18 

i£ 
30 

21 
22 
23 
24 
25 
26 
27 
28 
2i 
30 
31 
32 
33 
34 

35 
36 
37 



P.P. 





4^ 


43 


6 


4.2 


4.2 


7 


4^9 


4.9 


8 


5.6 


5.6 


9 


6.4 


6.3 


10 


7.1 


7.0 


20 


14.1 


14.0 


30 


21^2 


21.0 


40 


28^3 


28.0 


50 


35-4 


35. 





3f 


37 


6 


2.7 


2-7 


7 


3.2 


3.1 


8 


3^6 


3.6 


9 


4^1 


4.0 


10 


4.6 


4-5 


20 


9.1 


9^0 


30 


13.7 


13.5 


40 


18.3 


18.0 


50 


22.9 


22.5 





26 


6 


2.6 


7 


3.1 


8 


3.5 


9 


4.0 


10 


4-4 


20 


8.8 


30 


13.2 


40 


17.6 


50 


22.1 



36 

2.6 

3.0 

3-4 

3.9 

4.3 

8.6 

13-0 

17.3 

21.6 





2^ 


6 


2.5 


7 


3.0 


8 


3.4 


9 


3.8 


10 


4.2 


20 


8.5 


30 


12.7 


40 


17.0 


50 


21.2 



P.P. 



675 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
SS'' 53° 






9.58471 


1 


.58497 


2 


.58523 


3 


.58549 


4 


.58575 


5 


9.58601 


6 


.58626 


7 


.58652 


8 


.58678 


9 


.58704 


10 


9.58''30 


1] 


.58755 


12 


.58781 


13 


.58807 


14 


.58833 


15 


9.58859 


16 


.58884 


17 


.58910 


18 


.58936 


19. 


.58962 


30 


9.58987 


21 


.59013 


22 


.59039 


23 


.59064 


24 


.59090 



25 
26 
27 
28 
29. 
30 
31 
32 
33 
34 
35 
36 
37 



9.59116 
•59141 
.59167 
.59193 
.59218 



40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



Lg. Vers, 



9-59244 
^59270 
.59295 
.59321 
.59346 



9.59372 
.59397 
.59423 
. 59449 
•59474 



9.59500 
.59525 
.59551 
.595'76 
.59602 



•59627 
.59653 
.59678 
.59704 
.59729 



9. 



59754 
59780 
59805 
59831 
59856 



.59881 
.59907 
.59932 
.59958 
.59983 



9-60008 
' Lg.Vers, 




P.P. 





43 


43 


6 


4.2 


4-2 


7 


4.9 


4.9 


8 


5.6 


5.6 


9 


6.4 


6.3 


10 


7.1 


7.0 


20 


14.1 


14.0 


30 


21.2 


21.0 


40 


28.3 


28.0 


50 


35.4 


35.0 





36 


2^, 


6 


2.6 


2.S 


7 


3.0 


3.0 


8 


3.4 


3.4 


9 


3.9 


3.'i 


10 


4.3 


4.2 


20 


8.6 


8.5 


30 


13.0 


12.7 


40 


17-3 


17.0 


50 


21.6 


21.2 





35 


6 


2.5 


7 


2.9 


8 


3.3 


9 


3.7 


10 


4.1 


20 


8.3 


30 


12.5 


40 


16.6 


50 


20.8 



3¥^ 

2.3 

2.8 

3.2 

3.7 

4.1 

81 

12.2 

16.3 

20.4 



P.P. 



676 



TABLE Vllf.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTa 
54° 55° 



Lg. Versi 



Log.ExSi 



D 



Lg. Vers 



Log.Exs. 



P.P. 



61512 
61537 
61562 
61586 
61611 



61636 
61661 
61685 
61710 
61735 



61760 
61784 
61809 
61834 
61858 



61883 
61908 
61932 
61957 
61982 



62006 
62031 
62055 
62080 
62105 



62129 
62154 
62178 
62203 
62227 



62252 
62276 
62301 
62325 
62350 



62374 
62399 
62423 
62448 
62472 



62497 
62521 
62546 
62570 
62594 



62619 
62643 
62668 
62692 
62716 
62741 
62765 
62789 
62814 
62838 



62862 
62887 
62911 
62935 
62960 



9.62984 
Lg. Vers 



9. 



84590 
84632 
84675 
84717 
84759 



.84801 
•84843 
•84886 
84928 
.84970 



•85012 
•85054 
•85097 
•85139 
• ■85181 



•85223 
•85265 
•85308 
•85350 
•85392 



•85434 
•85476 
•85519 
•85561 
•85603 



•85645 
.85688 
•85730 
•85772 
•85814 



•85857 
.85899 
.85941 
.85983 
•86026 



9^ 



86068 
86110 
86152 
86195 
83237 




•86913 
•86956 
.86998 
.87040 
•87082 



9-87125 
-D Log.Exs 



9.62984 
.63008 
.63032 
.63057 
•63081 



•63105 
•63129 
.63154 
.63178 
•63202 



•63226 
•63250 
.63274 
•63209 
•63323 



9.63347 
•63371 
.63395 
•63419 
•63443 



9 •63468 
•63492 
.63516 
.63540 
•63564 



9 •63588 
.63612 
.63636 
.63660 
•63684 




9^63948 
.63972 
•63996 
.64019 
• 64043 



9 •64067 
.64091 
•64115 
.64139 
•64163 



9 •64187 
.64210 
.64234 
•64258 
•64282 



9 •64306 
.64330 
.64353 
.64377 
• 64401 



9-64425 
Lg. Vers 



24 
24 
24 
24 
24 
24 
24 
24 
24 

24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
-24 
24 
24 
24 
24 
24 
24 
24 
24 
23 
24 

24 
24 
24 
23 
24 

24 
23 
24 
24 
23 
24 
24 
23 
24 
23 
24 



9.87125 
.87167 
•87209 
.87252 
.87294 



9.87336 
•87379 
•87421 
•87463 
•87506 



9 •87548 
•87590 
.87633 
.87675 
•87717 



9^87760 
.87802 
.87844 
.87887 
•87929 



•87971 
•88014 
.88056 
•88099 
•88141 



•88183 
.88226 
.88268 
.88310 
.88353 



9.88395 
.88438 
.88480 
.88522 
•88565 



9^88607 
•88650 
.88692 
•88734 
•88777 




9^89244 
89286 
89329 
89371 
89414 



9.89456 
.89499 
.89541 
.89583 
•89626 



9-89668 
Log.Exs. 





1 
2 
3 

± 
5 
6 
7 
8 

_9 

10 

11 
12 
13 
Ik 
15 
16 
17 
18 
ii 
30 
21 
22 
23 
24 

25 
26 
27 
28 
29, 
30 
31 
32 
33 
34 
35 
36 
37 
38 
11 
40 
41 
42 
43 
44 

45 
46 
47 
48 
il 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59, 
60 





4^ 


43 


6 


4.2 


4.2 


7 


4.9 


4.9 


8 


5.6 


5.6 


9 


6.4 


6.3 


10 


7^1 


7.0 


20 


14^1 


14.0 


30 


21^2 


21.0 


40 


28^3 


28.0 


50 


35.4 


35.0 





35 


6 


2.5 


7 


2.9 


8 


3.3 


9 


3.7 


10 


4.1 


20 


8^3 


30 


12-5 


40 


16.6 


50 


20.8 



35- 

2.i 
2.8 
3.2 
3.7 

t\ 

12.2 
16.3 
20.4 





34 


35 


6 


2.4 


2.3 


7 


2.8 


2.7 


8 


3.2 


3.x 


9 


3.6 


3.5 


10 


4.0 


3.9 


20 


8.0 


7.5 


30 


12.0 


11.7 
15.6 


40 


16.0 


50 


20.0 


19.6 



P.P. 



677 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTSJ 
56° 57° 



Lg. Vers. 



O 

1 
2 
3 

5 
6 
7 
8 
JL 

10 

11 

12 
13 
14 
15 
16 
17 
18 
19 



9 . 64425 
. 64448 
•64472 
.64496 
•64520 



9 • 64543 
•64567 
•64591 
•64614 
.64638 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 



40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
60 
51 
52 
53 
54 
55 
56 
57 
58 
59 



.64662 
•64685 
•64709 
•64733 
•64756 



. 64780 
•64804 
•64827 
•64851 
.64875 



.64898 
•64922 
•64945 
•64969 
.64992 



.65016 
•65040 
•65063 
•65087 
.65110 



•65134 
•65157 
•65181 
•65204 
•65228 



.65251 
•65275 
•65298 
•65321 
.65345 



9.65368 
.65392 
•65415 
•65439 
.65462 

9.65485 
•65509 
•65532 
•65556 
•65579 



9.65602 
.65626 
.65649 
•65672 
.65696 



60 



9.65719 
•65742 
•65765 
•65789 
.65812 



9-65835 
Lg. Vers. 



Log.Exs, 




•90094 
•90136 
•90179 
•90221 
•90264 



•90306 
•90349 
•90391 
•90434 
•90476 



.90519 
•90561 
.90604 
•90647 
•90689 



.90732 
•90774 
•90817 
•90860 
•90902 



.90945 
•90987 
•91030 
•91073 
•91115 



•91158 
•91200 
•91243 
•91286 
•91328 



.91371 
•91414 
.91456 
.91499 
.91541 



•92011 
•92054 
.92096 
.92139 
•92182 
.92224 



-D Log.Exs 



1> 



Lg. Vers, 



9.65835 
•65859 
•65882 
.65905 
.65928 



9.65952 
.65975 
•65998 
.66021 
.66044 



•66068 
•66091 
.66114 
.66137 
.66160 



.66183 
.66207 
.66230 
.66253 
•66276 



•66322 
•66345 
.66368 
•66391 




9.66645 
.66668 
.66691 
•66714 
.66737 



9.66760 
.66783 
.66805 
.66828 
•66851 



9.66874 
.66897 
.66920 
.66943 
.66966 



9.66989 
.67012 
.67034 
.67057 
.67080 



9.67103 
.67126 
.67149 
.67171 
•67194 



9-67217 
Lg. Vers 



1> 



J> 



Log.Exs, 



.92224 
•92267 
.92310 
.92353 
•92395 



.92438 
.92481 
.92524 
.92566 
•92609 



•92652 
•92695 
.92737, 
.92780 
.92823 



•92866 
•92909 
•92951 
•92994 
•93037 



.93080 
•93123 
•93165 
•93208 
•93251 




•93722 
.93765 
•93808 
.93851 
.93894 



•93937 
•93980 
•94023 
.94066 
•94109 



9 •94151 
.94194 
.94237 
.94280 
•94323 



9.94366 
.94409 
.94452 
•94495 
.94538 



9.94581 
•94624 
•94667 
•94710 
.94753 



9-94796 



Log.Exs. 



2> 



JD 



P. P. 



6 

7 

8 

9 

10 

20 
30 
40 
50 



43 

4.3 



4SL 
4.2 
4 ■ 
5 



35-4 





24 


2J 


6 


2-4 


2. 


7 


2.8 


2- 


8 


3.2 


3. 


9 


3.6 


3. 


10 


4.0 


3. 


20 


8.0 


7. 


30 


12.0 


11. 


40 


16.0 


15. 


50 


20.0 


19. 



6 

7 

8 

9 

10 

20 

30 

40 

50 



23 


22 


2.3 


2.2 


2.7 


2 


6 


3.0 


3 





3.4 


3 


4 


3.8 


3 


7 


7.6 


7 




11.5 


11 


2 


15.3 


15 





19.1 


18 


7 



P.P. 



678 



c TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

58° 59° 



Ig, Vers 



67217 
67240 
67263 
67285 
67308 



67331 
67354 
67376 
67399 
67422 



67445 
67467 
67490 
67513 
67535 



67558 
67581 
67603 
67626 
67B4Q 



67671 
67694 
67717 
67739 
67762 



67784 
67807 
67830 
67852 
67875 



67897 
67920 
67942 
67965 
67987 



68010 
68032 
68055 
63077 
68100 



68122 
68145 
68167 
68190 
68212 



68235 
68257 
68280 
68302 
68324 



68347 
68369 
68392 
68414 
68436 



68459 
68431 
68503 
68526 
68548 



9.68571 



Lg. VerSi 



n 



Log.Exs, 



9.94796 
.94839 
•94882 
.94925 
•94968 



9.95011 
.95054 
•95097 
.95140 
.95183 



9.95226 
.95269 
.95313 
.95356 
•95399 



9.95442 
.95485 
.95528 
.95571 

•95614 




9^96089 
.96132 
.96175 
.96218 
.96261 



9-96305 
.96348 
.96391 
.96434 
•98478 



9^96521 
.96564 
.96607 
.96650 
.96694 



9.96737 
.96780 
.96824 
.96867 
•96910 



9.96953 
.96997 
.97040 
.97083 
.97127 



9.97170 
.97213 
.97257 
.97300 
^97343 

9-97387 



Log.Exs 



Ig. Vers 



•68571 
.68593 
.68615 
.68637 
•68660 



•68682 
.68704 
.68727 
.68749 
.68771 



•68793 
.68816 
.68838 
.68860 
.68882 



.68905 
.68927 
.68949 
.68971 
•68993 



.690] 6 
.69038 
.69060 
.69082 
•69104 



•69126 
.69149 
.69171 
.69193 
.69215 



9. 



69237 
69259 
69281 
69303 
69325 



9.69347 



.69392 
.69414 
.69436 



.69458 
.69480 
.69502 
.69524 
.69546 



.69568 
.69590 
.69612 
.69634 
•69656 



•69678 
.69700 
.69721 
.69743 
•69765 
.69787 
.69809 
•69831 
•69853 
.69875 



9-69897 



^ Lg. Vers 



I) 



Log. Exs. 



9.97387 
.97430 
.97473 
.97517 
.97560 



9-97603 
.97647 
.97690 
.97734 
.97777 



9.97820 
.97864 
.97907 
.97951 
•97994 



9^98038 
.98081 
.98125 
.98168 
•98211 



9-98255 
.98298 
.98342 
.98385 
.98429 



9.08472 
.98516 
.98559 
.98603 
•98647 



9.08690 
.98734 
.08777 
.98821 
.98864 



9.9R908 
.98952 
.98C95 
.99039 
•99082 



9-99126 
.99170 
.99213 
.99257 
.99300 



9.99344 
.99388 
.99431 
.99475 
-99519 




10-00000 



Log, Exs, 



I> 



P. P. 



6 


4-4 


7 


5.1 


8 


5-8 


9 


6-6 


10 


7-3 


20 


14-6 


30 


22-0 


40 


29-3 


50 


36.6 



43 

4.3 

5.1 

5.8 

6.5 

7.2 

14.5 

21.7 

29.0 

36.2 





43 


6 


4.3 


7 


5.0 


8 


5.7 


9 


6.4 


10 


7.1 


20 


14.3 


30 


21.5 


40 


28.6 


50 


35. B 





33 


6 


2.3 


7 


2.7 


8 


3.0 


9 


3-4 


10 


3-8 


20 


7-6 


30 


11-5 


40 


15-3 


50 


19.1 



22 

2.2 

2-6 

3-0 

3-4 

3-7 

7-5 

11-2 

15-0 

18.7 



23 


3l 


2.2 


2.: 


2-5 


2.5 


2-9 


2. 


3-3 


3.2 


3-6 


3.6 
7.1, 


7-.^ 


11-0 


10.7 


14-6 


14.3 


18.3 


17.9 



P. p. 



679 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, 
60'' 61** 

P.P. 




6 

7 

8 

9 

10 

20 

3C 

40 

50 



45 


41 




4.5 


4.4 




5.2 


5-2 




6.0 


5.9 




6.7 


6-7 




7.5 


7.4 




15.0 


14.8 




22.5 


22-2 




30.0 


29-6 




37.5 


37.1 





t 


44 


6 


4.4 


7 


5.1 


8 


5.8 


9 


6.6 


10 


7.3 


20 


14.6 


30 


22.0 


40 


29-3 


50 


36-6 



43 

4.3 

51 

5-8 

6.5 

7.2 

14-5 

21.7 

29.0 

36.2 





23 


31 


6 


2.2 


'J.l 


7 


2-5 


2-5 


8 


2.9 


2.8 


9 


3.3 


3.2 


10 


3.6 


3.e 


20 


7.3 


7.1 


30 


11.0 


10.7 


40 


14.6 


14.3 


50 


18.3 


17.9 



21 



6 


2.1 


7 


2.4 


8 


24 


9 


31 


10 


3.5 


20 


7.0 


30 


10.5 


40 


14.0 


50 


17.5 



P.P. 



680 



JABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
63° 63° 



Lg. Vers, 



9.72471 
72492 
72513 
72534 
72555 



72576 
72597 
72618 
72639 
72660 



72681 
72701 
72722 
72743 
72764 



72785 
72806 
72827 
72848 
72869 



72890 
72911 
72931 
72952 
72973 



72994 
73015 
73036 
73057 
73077 



73098 
73119 
73140 
73161 
73181 



73202 
73223 
73244 
73265 
73285 



73306 
73327 
73348 
73368 
73389 



73410 
73430 
73451 
73472 
73493 



73513 
73534 
73555 
73575 
73596 



73617 
73637 
73658 
73679 
73699 



9-73720 
|Lg.Vers 



I> Log, Exs, 



10.05310 
.05354 
.053 
•05444 
.05489 



10.05534 
.05579 
.05623 
.05668 
.05713 



10.05758 
.05803 
.05848 
.05893 
•05938 



10.05983 
.06028 
.06072 
.06117 
.06162 



10.06207 
.06252 
.06297 
.06342 
.06387 



10-06432 
.06477 
.06522 
.06568 
.06613 



10.06658 
.06703 
.06748 
.06793 
•06838 



10.06883 
•06928 
•06974 
•07019 
.07064 



10.07109 
•07154 
•07200 
.07245 
.07290 



10-07335 
•07380 
.07426 
.07471 
.07516 



10.07562 
.07607 
.07652 
.07697 
•07743 



10.07788 
.07834 
.07879 
.07924 
•07970 



10.08015 
Log, Exs 



2> 



Lg.Vers. 2) 



1.73720 
•73740 
.73761 
.73782 
.73802 




. 74028 
. 74049 
.74069 
•74090 
.74110 



9.74131 
.74151 
•74172 
•74192 
•74213 



9.74233 
.74254 
.74274 
.74294 
•74315 



1.74335 
•74356 
•74376 
•7439& 
•74417 



• 74437 
•74458 
•74478 
•74498 
.74519 



.74539 
.74559 
•74580 
.74600 
•74620 



. 74641 
•74661 
•74681 
•74702 
•74722 



9 . 74742 
•74762 
•74783 
•74803 
.74823 



1.74844 
•74864 
.7488^ 
.74904; 
-74924 



9-74945 



Lg. Vers 



JD 



Log, Exs. 



10-08015 
.08061 
.08106 
.08151 
•08197 




10^08697 
.08743 
.08789 
.08834 
•08880 



10-08926 
.08971 
.09017 
.09062 
.09108 



10.09154 
.09200 
.09245 
•09291 
.09337 



10-09382 
•09428 
•09474 
•09520 
.09566 



10.09611 
.09657 
.09703 
.09749 
•09795 




10.10300 
•10346 
•10392 
•10438 
.10484 



10-10530 
•10576 
.10622 
.10668 
•10714 



10.10760 



Log. Exs. 



JD 



P.P. 





46 


6 


4.6 


7 


5.4 


8 


6.2 


9 


7.0 


10 


7.7 


20 


15.5 


30 


23.2 


40 


31.0 


50 


38.7 



6 

7 

8 

9 

10 

20 

30 

40 

50 



45 



4-5 


4. 


5-3 


5. 


6.0 


6- 


6^8 


6. 


7.6 


7- 


15.1 


15. 


22.7 


22- 


30^3 


30. 


37.9 


37. 



46 

4-6 

5.3 

6-1 

6.9 

7-6 

15.3 

23.0 

30.6 

38.3 



45 
5 

2 

7 
5 

5 

5 



6 
7 
8 

9 

10 
20 
30 
40 
50 



4? 

4-4 

§•2 

§•9 

6.7 

7.4 

14.8 

22-2 

29-6 

37-1 





31 


6 


2.1 


7 


2.4 


8 


2.8 


9 


3.1 


10 


3.5 


20 


7-0 


30 


10.5 


40 


14.0 


50 


17.5 



6 
7 
8 
9 

10 
20 
30 
40 
50 



30 

2-0 

24 

2.7 

31 

34 

6-8 

10-2 

13-6 

17.1 



30 

2.0 



2.6 
3.0 
3-3 

6-6 
10-0 
13-5 
16.6 



P. P. 



681 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECAN^ 
64° 65° 



25 
26 
27 
28 
29 
30 
31 
32 
33 
34 



40 

41 
42 
43 
44 



.Vers. 2> 






9-74945 


1 


•74965 


2 


•74985 


3 


-75005 


4 


-75026 


5 


9^75046 


6 


.75066 


7 


•75086 


8 


•75106 


9 


•75126 


10 


9-75147 


11 


•75167 


12 


•75187 


13 


•75207 


14 


-75227 


15 


9-75247 


16 


•75267 


17 


•75287 


18 


•75308 


19 


-75328 


20 


9-75348 


21 


•75368 


22 


•75388 


23 


•75408 


24 


•75428 




9-75648 
•75668 
•75688 
•75708 
-75728 



9-75748 
•75768 
.75788 
-75808 
-75828 



45 
46 
47 
48 
49 
50 
51 
52 
53 
51 
55 
56 
57 
58 
59 



9-75848 
-75868 
-75888 
-75908 
■75928 



60 



9-75947 
75967 
75987 
76007 
76027 



9-76047 
76067 
76087 
76106 
76126 



9-76146 



Log.Exs. 



10. 



10760 
10807 
10853 
10899 
10945 



10. 



10991 
11037 
11084 
11130 
11176 



10. 



11222 
11269 
11315 
11361 
11407 



10- 



11454 
11500 
11546 
11593 
11639 



10. 



11685 
11732 
11778 
11825 
11871 



10- 



11917 
11964 
12010 
12057 
12103 



10 



12150 
12196 
12243 
.12289 
12336 



10 



12383 
12429 
12476 
12522 
12569 



10. 



10 



12616 
12662 
12709 
12756 
12802 

12849 
12896 
12942 
12989 
13036 



10 



' Lg. Vers. 



13083 
13130 
.13176 
.13223 
•13270 



D 



10. 



13317 
13364 
13411 
13457 
13504 



10-13551 

J> Log.Exs. 



Lg. Vers, 



76146 
76166 
76186 
76206 
76225 



76245 
76265 
76285 
76304 
76324 



76344 
76364 
76384 
76403 
76423 



76443 
76463 
76482 
76502 
76522 



76541 
76561 
76581 
76600 
76620 



76640 
76659 
76679 
76699 
76718 



76738 
76758 
76777 
76797 
76817 



76836 
76856 
76875 
76895 
76915 



76934 
76954 
76973 
76993 
77012 



I> Log.Exs, 



77032 
77052 
77071 
77091 
77110 



77130 
77149 
77169 
77188 
77208 



77227 
77247 
77266 
77286 
77305 



77325 



1> Lg.Vers 



10-13551 
•13598 
.13645 
.13692 
•13739 



10.13786 
.13833 
.13880 
.13927 
•13974 



10.14021 
•14068 
•14115 
.14162 
-14210 



10-14257 
•14304 
•14351 
.14398 
- 14445 



10-14493 
.14540 
.14587 
.14634 
•14682 



10^14729 
.14776 
.14823 
.14871 
.14918 



10.14965 
.15013 
.15060 
.15108 
.15155 



D 



10.15202 
.15250 
.15297 
.15345 
.15392 



10.15440 
.15487 
.15535 
.15582 
.15630 



10.15678 
.15725 
.15773 
.15820 
•15868 



10^15916 
.15963 
.16011 
.16059 
•16106 



10^16154 
•16202 
.16250 
.16298 
.16345 



10.16393 
Log. Exs. 



47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 

47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 
47 

4f 

47 
47 
47 
4Z 
47 
47 
47 

^1 
41 

47 
47 
48 

47 
47 
48 
47 
47 
48 
47 
48 
48 
47 
48 



o 

1 

2 
3 

_4 
5 
6 
7 
8 

_9 
10 
11 
12 
13 
Ik 
15 
16 
17 
18 
19. 
30 
21 
22 
23 
24 
25 
26 
27 
28 
li 
30 
31 



P.P. 



48 



4 


8 


5 


6 


6 


4 


7 


2 


8 





16 





24 





32 





40 






47 

4-7 

5^5 

6^3 

7-1 

7-9 

15^8 

23^7 

31^6 

39.6 





47 


46 


6 


4.7 


4.6 


7 


5.5 


5 


4 


8 


6^2 


6 


2 


9 


7-0 


7 





10 


7-8 


7 


7 


20 


15^6 


15 




30 


23^5 


23 


2 


40 


31-3 


31 





50 


39.1 


38 


7 



46 

4-6 



30 23 
4030 
50138 



61 

7 

8 

9 

10 

20 

30 

40 

50 



30 

2-0 

'•4 

.••7 

3-1 

3-4 

6-8 

10-2 

13-6 

17.1 



20 

2-0 



6 

7 

8 

9 

10 

20 

30 

40 

50 



19 

1-9 
2-3 
2.6 
2-9 
3-2 
6.5 
9^7 
13^0 
16.2 



P.P. 



682 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
66° 67° 



' L 


g.Vers. 


s 

1 

2 
3 
4 


.77325 
.77344 
.77363 
.77383 
•77402 


5 S 

6 

7 

8 

9 


.77422 
.77441 
.77461 
.77480 
.77499 


10 S 

11 
12 
13 
14 

15 9 

16 

17 

18 

19 


.77519 
.77538 
.77557 
.77577 
.77596 

.77616 
.77635 
.77654 
.77674 
.77693 


30 9 

21 

22 

23 

24 


.77712 
.77732 
.77751 
.77770 
.77790 


25 9 

26 

27 

28 

29 


.77809 
.77828 
.77847 
.77867 
.77886 


30 9 

31 
32 
33 
34 


.77905 
.77925 
.77944 
.77963 
.77982 


35 9 

36 

37 

38 

39 


.78002 
.78021 
.78040 
.78059 
•78078 


40 c 

41 
42 
43 
44 


•78098 
.78117 
.78136 
.78155 
•78174 


45 c 

46 

47 

48 

49 


•78194 
.78213 
.78232 
.78251 
•78270 


50 c 

51 
52 
53 
54 

55 J 

56 

57 

58 

59_ 

60( 


1.78289 
.78309 
.78328 
.78347 
•78366 


). 78385 
.78404 
.78423 
.78442 
.78462 


). 78481 



' Lg. Vers 



JD 



Log.Exs. 



10.16393 
.16441 
.16489 
.16537 
.16585 



10.16633 
.16680 
.16728 
.16776 
.16824 



10.16872 
.16920 
.16968 
.17016 
.17064 



10-17112 
.17160 
.17209 
.17257 
.17305 



10.17353 
.17401 
.17449 
.17498 
.17546 



10.17594 
.17642 
.17690 
.17739 
.17787 



10.17835 
.17884 
.17932 
.17980 
.18029 



10.18077 
.18126 
.18174 
.18222 
.18271 



10.18319 
.18368 
.18416 
.18465 
.18514 



10.18562 
.18611 
.18659 
.18708 
.18757 



10.18805 
.18854 
.18903 
.18951 
.19000 



10.19049 
.19098 
.19146 
.19195 
.19244 



10-19293 
Log.Exs, 



n 



Lg. Vers, 



9.78481 
.78500 
.78519 
.78538 
.78557 



9-78576 
.78595 
.78614 
.78633 
•78652 



9-78671 
.78690 
.78709 
.78728 
.78747 



9.78766 
.78785 
.78804 
.78823 
.78842 



9.78861 
.78880 
.78899 
.78918 
=78937 



9.78956 
.78975 
.78994 
.79013 
.79032 



9.79051 
.79069 
.79088 
.79107 
.79126 



9.79145 
.79164 
.79183 
.79202 
.79220 




9.79427 
.79446 
.79465 
.79484 
.79503 



9.79521 
.79540 
.79559 
.79578 
.79596 



9-79615 
Lg. Vers 



Log.Exs. 



10. 



10. 



19293 
19342 
19391 
19439 
19488 

19537 
19586 
19635 
19684 
19733 



10- 



19782 
19831 
19880 
19929 
19979 



10. 



20028 
20077 
20126 
20175 
20224 



10. 



20273 
20323 
20372 
20421 
20470 



10 



30520 
20569 
.20618 
20668 
20717 



10 



10 



20767 
20816 
20865 
20915 
.20964 
21014 
21063 
.21113 
.21162 
.21212 



10- 



21262 
21311 
21361 
21410 
21460 



10 



21510 
21560 
21609 
21659 
21709 



10 



21759 
21808 
.21858 
21908 
.21958 



10. 



22008 
22058 
22108 
22158 
22208 



10-22258 

-O Log.Exs. 



1> 



P.P. 





50 


6 


5.0 


7 


5-8 


8 


6-6 


9 


7.5 


10 


8-3 


20 


16-6 


30 


25.0 


40 


33.3 


50 


41.6 





49 


6 


4.9 


7 


5.7 


8 


6-5 


9 


7-3 


10 


8.1 


20 


16-3 


30 


24-5 


40 


32.6 


50 


40.8 





48 


6 


4.8 


7 


5-6 


8 


6-4 


9 


7.2 


10 


8-0 


20 


16-0 


30 


24-0 


40 


32.0 


50 


40.0 





19 


6 


1.9 


7 


2.3 


8 


2.6 


9 


2.9 


10 


3.2 


20 


6.5 


30 


9.7 


40 


13-0 


50 


16.2 



10 
20 
30 
40 
50 



49 

4.9 

5.8 

6.6 

7.4 

8.2 

16.5 

24.7 

33-0 

41-2 



48 

4.8 

5.6 

6.4 

7.3 

8.1 

16.1 

24.2 

32.3 

40.4 



47« 

n 

6.3 
7.1 

15^5 
23.7 
31-6 
39-6 



19 

1.9 

2*5 

2.e 

3.1 

6.3 

9.5 

12.6 

15. § 



1?. 

1.8 
2.1 
2.4 
2.8 
3.1 
6.1 
9.2 
12.3 
15.4 



P.P. 



683 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 
68° 69° 



Lg, Vers, 



79615 
79634 
79653 
79671 
79690 



79709 
79727 
79746 
79765 
79783 



79802 
79821 
79839 
79858 
79877 



79895 
79914 
79933 
79951 
79970 



79988 
80007 
80026 
80044 
80063 



80081 
80100 
80119 
80137 
80156 



80174 
80193 
80211 
80230 
80248 



80267 
80286 
80304 
80323 
80341 



80360 
80378 
80397 
80415 
80434 



80452 
80470 
80489 
80507 
80526 



80544 
80563 
80587 
80600 
80618 



80636 
80655 
80673 
80692 
80710 



80728 
Lg. Vers 



2> 



10.23262 
.23312 
.23362 
.23413 
.23463 



Log.Exs. 



10-22258 
.22308 
.22358 
.22408 
.22458 



10-22508 
.22558 
.22608 
.22658 
.22708 



10.22759 
.22809 
.22859 
.22909 
.22960 



10.23010 
.23060 
.23110 
.23161 
.23211 



10.23514 
.23564 
.23615 
.23666 
.23716 



10.23767 
.23817 
.23868 
.23919 
.23969 



10.24020 
.24071 
.24122 
.24172 
.24223 



10.24274 
.24325 
.24376 
.24427 
.24478 



10.24529 
.24580 
.24631 
.24682 
.24733 



10.24784 
.24835 
.24886 
.24937 
.24988 



10.25039 
.25095 
.25142 
.25193 
.25244 



10.25295 



Log.E; 



xs. 



Lg. Vers, 



80728 
80747 
80765 
80783 
80802 



80820 
80839 
80857 
80875 
80894 



80912 
80930 
80949 
80967 
80985 



81003 
81022 
81040 
81058 
81077 



81095 
81113 
81131 
81150 
81168 



81186 
81204 
81223 
81241 
81259 



81277 
81295 
81314 
81332 
81350 



81368 
81386 
81405 
81423 
81441 



81459 
81477 
81495 
8151" 
81532 



81550 
81568 
81586 
81604 
81622 



81640 
81658 
81676 
81695 
81713 



81731 
81749 
81767 
81785 
81803 
9.81821 
Lg. Vers. 



Log.Exs. 



10 



25295 
25347 
25398 
25449 
25501 



10. 



25552 
25604 
25655 
25707 
25758 



10 



25810 
25861 
25913 
25964 
2S016 



10. 



26067 
26119 
26171 
26222 
26274 



10 



26326 
26378 
26429 
26481 
26533 



10. 



26585 
26637 
26689 
26741 
26793 



10. 



26845 
26897 
26949 
27001 
27053 



10. 



27105 
27157 
27209 
27261 
27314 



10.27366 
.27418 
•27470 
.27523 
■27575 



10-22627 
.27680 
.27732 
.27785 
.27837 



10.27890 
.27942 
.27995 
.28047 
■28100 



10-28152 
.28205 
.28258 
.28310 
.28363 



I> 



10-28416 



Log.Exs. 



10 

11 
12 
13 
14 

15 
16 
17 
18 
19 
30 
21 
22 
23 
2^ 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 
47 
48 
_49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59, 
60 



P.P. 





53 


53 


6 


5-3 


5-2 


7 


6.2 


6 


1 


8 


7-0 


7 





9 


7-9 


7 


9 


10 


8-8 


8 


7 


20 


17-6 


17 


5 


30 


26.5 


26 


2 


40 


35.3 


35 





50 


44.1 


43 


7 



6 

7 

8 

9 

10 

20 

30 

40 

50 



53 


5 


2 


6 





6 


9 


7 


8 


8 


6 


17 


3 


26 





34 


(i 


43 


3 



6 


5.1 


7 


5.9 


8 


6.8 


9 


7.6 


10 


8.5 


20 


17.0 


30 


25.5 


40 


34.0 


50 


42.5 



51 

5.1 

6-0 

6-8 

7.7 

8-6 

17.1 

25-7 

34-3 

42.9 

50 

5-0 

5.9 

6-7 

7.6 

8-4 

16-8 

25-2 

33-6 

42.1 



50 

50 

5.8 

66 

7.5 

8.3 

16.6 

25.0 

33.3 

41.6 



6 

7 
8 
9 

10 
20 
30 
40 
50 



19 

1.9 
2.2 
2.5 
2-8 
31 
6.3 
9-5 
12-6 
15.8 



18 

1.8 



6 


1 


7 


2. 


8 


2- 


9 


2- 


10 


3. 


20 


6. 


80 


9- 


40 


12. 


50 


15. 



18 

-8 
.1 
-4 
7 


-0 

.0 



p.p. 



684 



/ABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 



Lg.Vers. 



9.81821 
81839 
81857 
81875 
81893 



81911 
81929 
81947 
81965 
81983 

82001 
82019 
82037 
82055 
82073 



82091 
82109 
82127 
82145 
82163 
82181 
82199 
82217 
82235 
82252 
82270 
82288 
82306 
82324 
82342 



82360 
82378 
82396 
82413 
82431 



82449 
82467 
82485 
82503 
82520 



82538 
82556 
82574 
82592 
82609 



82627 
82645 
82663 
82681 
82698 



82716 
82734 
82752 
82769 
82787 



82805 
82823 
82840 
82858 
82876 



9.82894 



Lg, Vers. 



D 



10.28945 
.28998 
•29051 
.29104 
.29157 

10.29210 
.29263 
.29316 
.29370 
.29423 



10.29476 
.29529 
.29583 
.29636 
.29689 



J> 



Log. Exs, 



10.28416 
.28469 
.28521 
.28574 
.28627 



10-28680 
.28733 
.28786 
.28839 
.28892 




10.30278 
.30332 



.30440 
,.30493 



10.30547 
.30601 
.30655 
.30709 
.30763 



10.30817 
.30871 
.30925 
.30979 
.31033 



10.31087 
.31141 
•31195 
•31249 
.31303 



10.31358 

•31412 
•31466 
.31521 
.31575 



I) 



10-31629 



Log. Exs. J J> Lg. Vers, 



Lg.Vers. I> Log. Exs 



.82894 
•82911 
• 829*59 
•82947 
■82964 



.82982 
•83000 
•83017 
.83035 
.83053 



•83070 
•83088 
.83106 
.83123 
•83141 



9. 



83159 
83176 
83194 
83211 
83229 



.83247 
•83264 
•83282 
•83299 
•83317 



•83335 
•83352 
•83370 
•83387 
.83405 




.83598 
.83615 
•83633 
.83650 
.83668 



.83685 
.83703 
•83720 
.83737 
•83755 



•83772 
.83790 
.83807 
.83825 
•83842 




10-31629 
.31684 
.31738 
•31793 
•3184 7 

10-31902 
•31956 
•32011 
•32066 
.32120 



10.32175 
.32230 
.32284 
•32339 
•32394 



10.32449 
•32504 
•32558 
•32613 



10-32723 
•32778 
•32833 
•32888 
.32944 




10-33552 
•33607 



•33718 
.33774 



10.33829 
.33885 
•33941 
.33996 
•34052 



10.34108 
•34164 
.34220 
-34275 
•34331 



10 •34387 
•34443 
•34499 
•34555 
•34611 



10-34667 
-34723 
-34780 
•34836 
•34892 



10-34948 



Log. ExS: 



2> 



J> 



O 

1 
2 
3 

4 

5 
6 
7 
8 
__9 
10 
11 
12 
13 
JA 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29. 
30 
31 
32 
33 
34 



P.P. 



6 

7 
8 
9 
10 
20 
30 
40 
50, 



56_ 

5.6 

6^6 

7^5 

8^5 

9^4 

18.8 

28.2 

37.6 

47.1 





55 


6 


5.5 


7 


6.5 


8 


7.4 


9 


8.3 


10 


9.2 


20 


18.5 


30 


27.7 


40 


37.0 


50 


46.2 



6 
7 
8 

9 

10 
20 
30 
40 
50 



5^ 

5.4 

6-3 

7^2 

8.2 

9.1 

18.1 

27.2 

36-3 

45-4 





53 


6 


5.3 


7 


6.2 


8 


?•! 


9 


8.0 


10 


8.9 


20 


17.8 


30 


26.7 


40 


35.6 


50 


44-6 



56 

5.6 

6-5 

7-4 

8-4 

9-3 

18.6 

28-0 

37.3 

46.6 

55 

5.5 

6.4 

7-3 

8.2 

9.1 

18-3 

27-5 

36-6 

45-8 

54 

5.4 

6-3 

7-2 

8^1 

9^0 

18^0 

27-0 

36-0 

45-0 

53 

5-3 

6-2 

7-0 

7^9 

8^8 

17^6 

26-5 

35-3 

44.1 



6 
7 
8 

9 
10 
20 
30 
40 
50 



53 

5-2 

6.1 

7.0 

7.9 

8.7 

17.5 

26.2 

35.0 

43-7 





18 


17 


6 


1-8 


1.7 


7 


2-1 


2-0 


8 


2-4 


2^3 


9 


2.7 


2.6 


10 


3.0 


2.9 


20 


6.0 


5.8 


30 


9.0 


8.7 


40 


12.0 


11-6 


50 


15-0 


14-6 



17 

1.7 
2.0 
2.2 
2.5 
2-8 
5-6 
8-5 
11-3 
14-1 



P.P. 



685 



TABLE VIII.— LOGARITHMIC VERSED SINES AND ^TERNAL SECANTS 



Lg. Vers 



83946 
83964 
83981 
83999 
84016 



84033 
84051 
84068 
84085 
84103 



84120 
84137 
84155 
84172 
84189 



84207 
84224 
84241 
84259 
84276 



84293 
84310 
84328 
84345 
84362 



84380 
84397 

84414 
84431 
84449 




84638 
84655 
84672 
84689 
84706 



84724 
84741 
84758 
84775 
84792 



84809 
84826 
84844 
84861 
84878 



84895 
84912 
84929 
84946 
8 4963 
84980 



J> 



Log.Exs. 



10.34948 
.35005 
•35061 
.35117 
.35174 



10-35230 
.35286 
.35343 
.35399 
.35456 



10.35513 
.35569 
.35626 
.35683 
.35739 



10.35796 
.35853 
.35910 
.35967 
.36023 



10.36080 
.36137 
.36194 
.36251 
.36308 



10.36366 
.36423 
.36480 
.36537 
.36594 



10.36652 
.36709 
.36766 
.36824 
.36881 



10.36938 
•36996 
•37054 
•37111 
.37169 




10.37515 
•37573 
.37631 
.37689 
.37747 




J> 



Lg. Vers 




84980 
84997 
85014 
85031 
85049 



85236 
85253 
85270 
85287 
85304 



85321 
85338 
85355 
85372 
85389 




85575 
85592 
85608 
85625 
85642 



85659 
85676 
85693 
85710 
85726 




85911 
85928 
85945 
85962 
85979 



985995 



D 



Log. Exs. 



10. 



38387 
38445 
38504 
38562 
38621 



10. 



38679 
38738 
38796 
38855 
38914 



10. 



38973 
39031 
39090 
39149 
39208 



10. 



10. 



39267 
39326 
39385 
39444 
39503 
39562 
39621 
39681 
39740 
39799 



10 



39859 
39918 
39977 
40037 
40096 



10. 



40156 
40216 
40275 
40335 
40395 



10 



40454 
40514 
40574 
40634 
40694 



10 



40754 
40814 
40874 
40934 
40994 



10 



41054 
41114 
41174 
41235 
41295 



10 



41355- 
41416 
41476 
41537 
41597 



10 



41658 
41719 
41779 
41840 
41901 



10-41962 



1> 



O 

1 

2 
3 
± 
5 
6 
7 
8 
9. 

10 

11 
12 
13 
li 
15 
16 
17 
18 
11 

30 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 
42 
43 
44 

45 
46 
47 
48 
ii 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 

60 



P. P. 





61 


6 


6.1 


7 


7.1 


8 


8. 


9 


9.1 


10 


10. 


20 


20.3 


30 


30.5 


40 


40.6 


50 


50.8 





60 


6 


6.0 


7 


7.0 


8 


8.0 


9 


9.0 


10 


10.0 


20 


20.0 


30 


30.0 


40 


40.0 


50 


50.0 



60 

6.0 
7.0 
8.0 
9.1 

10.1 
20.1 
30.2 
40.3 
50.4 

59 

5.9 

6.9 

7.9 

8.9 

9.9 

19.8 

29.7 

39.6 

49.6 





59 


58j 


6 


5.9 


5.8 


7 


6.9 


6.8 


8 


7.8 


7.8 


9 


8.8 


88 


10 


9.8 


9.7 


20 


19.6 


19.5 


30 


29.5 


29.2 


40 


39.3 


39.0 


50 


49.1 


48.7 





58 


6 


5.8 


7 


6.7 


8 


7.7 


9 


8.7 


10 


9.6 


20 


19.3 


30 


29-0 


40 


38.6 


50 


48.3 



57 

5.7 
6.7 
7.6 
8.6 

lU 

28.7 
38.3 
47.9 





57 


56_ 


6 


5.7 


5.6 


7 


6.6 


6.6 


9 


7.6 


7.5 


9 


8.5 


8.5 


10 


9.5 


9.4 


20 


19.0 


18.8 


30 


28-5 


28.2 


40 


38.0 


37.6 


50 


47.5 


47.1 





17 


17 


6 


1.7 


1.7 


7 


2.0 


2-0 


8 


2.3 


2.2 


G 


2.6 


2.5 


10 


2.9 


2.8 


20 


5.8 


5.6 


30 


8.7 


8.5 


40 


11.6 


11.3 


50 


14.6 


14.] 



16 

1.6 
1.9 
2.2 
2.5 
2.7 
5.5 
8.2 
11.0 
13.7 



I 



Lg. Vers, 



JJ 



Log.Exs. 



Lg. Vers, 



^ Log. Exs. 
686 



P. P. 



TABLE VIII.— LOGAHITHMIC VERSED SINES AND EXTERNAL SECANTS* 
74° 75° 






9.85995 


1 


.86012 


2 


.86029 


3 


.86046 


4 


.86062 


5 


9.86079 


6 


.86096 


7 


.86113 


8 


.86129 


9 


-.86146 




39, 
40 

41 
42 
43 
44 

45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



P.P. 



67 


66 


6.7 


6.6 


7.8 


7.7 


8.9 


8.8 


10.0 


10.0 


11.1 


11.1 


22.3 


22.1 


33-5 


33.2 


44.6 


44.3 


55.8 


55.4 



66 

6.6 
7-7 
8.8 

9.9 
110 
22.0 
330 
44.0 
55-0 



65 

6.5 

7.6 

8.7 

9.8 

10.9 

21.8 

32.7 

43.6 

54-6 



65 

6.5 

7.6 

8.6 

9.7 
10.8 
21.6 
32.5 
43.3:43 
54.li53 



64 





64 


63 


6 


6-4 


6.3 


7 


7.4 


7.4 


8 


8.5 


8.4 


9 


9.6 


9.5 


10 


10.6 


10.6 


20 


21.3 


21.1 


30 


32.0 


31.7 


40 


42.6 


42.3 


50 


53.3 


52.9 





62 


62 


6 


6.2 


6.2 


7 


7.3 


7.2 


8 


8.3 


8.2 


9 


9.4 


9.3 


10 


10.4 


10-3 


20 


20.8 


20.6 


30 


31.2 


310 


40 


41.6 


41.3 


50 


52.1 


51.6 



63 

6.3 

7.3 

8.4 

9.4 

10.5 

21.0 

31-5 

42 

52.5 

61. 

6.1 

7.2 

8.2 

9.2 

10.2 

20.5 

30.7 

41.0 

51.2 



7 
8 
9 
10 
20 
30 
40 
50 



61 


6 


1 


7 




8 




9 




10 




20 


3 


30 


5 


40 


6 


50 


81 



60 

6.0 

7.0 

8.0 

9.1 

10.1 

20.1 

30.2 

40.3 

50.4 



17 

1.7 
2.0 
2.2 
2.5 
2.8 
5.6 
8.5 
11.3 
14.1 



16 

1-6 
l.J 
2.2 
2. J 

2.': 

5.5 

8.2 

11.0 

13.7 

P. P. 



16 

1.6 
1.8 
2.1 

2.4 
2.6 
5.3 
8.0 
10.6 
13.3 



687 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS 



5 
6 
7 
8 

10 

11 
12 
13 
li 
15 
16 
17 
18 
19 



9.88052 
.88068 
.88084 
.88100 
.88116 



30 

21 
22 
23 
24 
25 
26 
27 
28 
29 



80 
81 

32 
33 
31 
35 
36 
37 
38 



40 

41 
42 
43 
44 

45 
46 
47 
48 
49 



Lg.Vers. 



9.87971 
.87987 
.88003 
.88020 
.88036 



9-88133 
.88149 
.88165 
.88181 
.88197 



9.88213 
.88229 
.88245 
.88261 
■88277 



.88294 
.88310 
.88326 
.88342 
.88358 



.88374 
.88390 
.88406 
.88422 
^88438 



.88454 
.88470 
.88486 
.88502 
.88518 



.88534 
.88550 
.88566 
.88582 
■88598 



.88614 
.88630 
.88646 
.88662 
.88678 



9.88694 
.88710 
.88726 
.88742 
.88758 



50 9.88774 



51 
52 
53 
54 

55 
56 
57 
58 
59 
60 



.88790 
88805 
88821 
88837 



9 88853 

.88869 

.88885 

■88901 

_JB8917 

9-88933 



1> 



Log.Exs. 



10 



49604 
49670 
49737 
49804 
49871 



10 



49939 
50006 
50073 
50140 
50208 



10 



.50275 
.50342 
.50410 
.50477 
.50545 



10. 



50613 
50681 
50748 
50816 
50884 



10. 



50952 
51020 
51088 
51157 
51225 



10. 



51293 
51361 
51430 
51498 
51567 



10. 



51636 
51704 
51773 
51842 
51911 



10. 



51980 
52049 
52118 
52187 
52256 



10. 



52325 
52394 
52464 
52533 
52603 



10. 



52672 
52742 
52812 
52881 
529GI 



10. 



53021 
53091 
5316T 
53231 
5330T 



10. 



53372 
53442 
53512 
53583 
53653 



10 53724 



69 



Lg, Vers. 



■8893^ 
.88949 
.88964 
.88980 
•88996 



.89012 
.89028 
.89044 
.89060 
.89075 




.89249 
.89265 
.89281 
.89297 
.89312 



.89328 
.89344 
.89360 
.89376 
.89391 



.89407 
.89423 
.89438 
.89454 
■89470 



■89486 
.89501 
.89517 
.89533 
.89548 



.89564 
.89580 
•89596 
.89611 
.89627 



. 89643 
.89658 
.89674 
.89690 
.89705 



.89721 
.89737 
.89752 
.89768 
.89783 
■89799 
.89815 
.89830 
.89846 
•89862 



9 89877 



D 



Log.Exs. 



10.53724 
.53794 
.53865 
.53936 
•54007 



10.54078 
. 54149 
.54220 
.54291 
•54362 



10.54433 
.54505 
.54576 
. 54647 
.54719 



10.54791 
.54862 
.54934 
.55006 
.55078 



10.55150 
.55222 
.55294 
.55366 
.55438 



10.55511 
•55583 
•55655 
•55728 

^5801 



10.55873 
•55946 
•56019 
•56092 
.56165 



10.56238 
.56311 
.56384 
.56457 
.56531 



10.56604 
•56678 
•56751 
.56825 
.56899 



10.56973 
•57047 
.57120 
•57195 
•57269 



10.57343 
•57417 
•57491 
•57566 
.57640 



10.57715 
.57790 
.57864 
•57939 
•58014 



10.58089 



D 



10 

11 
12 
13 
JA 
15 
16 
17 
18 
19 

30 

21 
22 
23 
_24 
25 
26 
27 
28 

30 

31 
32 
33 

3A 
35 
36 
37 
38 
39 

40 
41 
42 
43 

j44 

45 
46 
47 
48 
49 

50 
51 
52 
53 

_34 

55 
56 
57 
58 

60 



P.P. 





75 


74 


6 


7.5 


7.4 


7 


8.7 


8^6 


8 


10.0 


9.8 


9 


11.2 


11.1 


10 


12.5 


12.3 


20 


25.0 


24.6 


30 


37.5 


37.0 


40 


50.0 


49.3 


50 


62.5 


61.6 





73 


71 


6 


7.2 


7.1 


7 


8.4 


8.3 


8 


9.6 


9.4 


9 


10.8 


10^6 


10 


12.0 


11.8 


20 


24.0 


23-6 


30 


36.0 


35^5 


40 


48.0 


47^3 


50 


60.0 


59.1 



73 

7.3 
8.5 
9.7 
10.9 
12.1 
24.3 
36.5 
48.6 
60.8 



70 

7.0 
8.2 
9.4 
10.6 
11.7 
23.3 
35-2 
47^0 
58.7 



69 


68 


67 


6 6.9 


6.8 


6.7 


7 8.0 


7.9 


7.8 


8 9.2 


9.0 


8.9 


9 10.3 


10.2 


10.0 


10^11.5 


11.3 


11.1 


2023.0 


22.6 


22.3 


3034.5 


34.0 


33.5 


4046.0 


45-3 


44.6 


50!57.5 


56.6 


55. S 





66 





6 


6.6 


0.0 


7 


7.7 


o.g 


8 


8.8 


0.0 


9 


9-9 


0.1 


10 


11.0 


0.1 


20 


22.0 


O.I 


30 


33.0 


0.2 


40 


44.0 


0.3 


50 


55.0 


0.4 





16 


16 


15 


6 


1.6 


1.6 


1^5 


7 


1.9 


1 


8 


1^8 


8 


2.2 


2 


1 


2^0 


9 


2^5 


2 


4 


2^3 


10 


2.7 


2 


6 


2^6 


20 


5.5 


5 


3 


51 


30 


8.2 


8 


-0 


7.7 


40 


11.0 


10 


6 


10.3 


50 


13.7 


13 


3 


12.9 



Lg.Vers 



-O Log.Exs. 



J>|Lg.Vers. 



X> 



Log.Exs. 



2> 



P.P. 



TABLE VIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
78° 79° 



Lg. Vers. I> Log.Exs. I> Lg. Vers. I> 



89877 
89893 
89908 
89924 
89939 



89955 
89971 
89986 
90002 
90017 



90033 
90048 
90064 
90080 
90095 



90111 
90126 
90142 
90157 
90173 



90188 
90204 
90219 
90235 
90250 



90266 
90281 
90297 
90312 
90328 



90343 
90359 
90374 
90389 
90405 



90420 
90436 
90451 
90467 
90482 



90497 
90513 
90528 
90544 
90559 



90574 
90590 
90605 
90621 
90636 



90651 
90667 
90682 
90697 
90713 



90728 
90744 
90759 
90774 
90790 



90805 



' Lg. Vers. 



10.58089 
.58164 
.58239 
.58315 

^8390 



10.58465 
.58541 
.58616 
.58692 
.58768 



10.58844 
. 58920 
.58995 
.59072 
.59148 



10.59224 
.59300 
.59377 
.59453 
.59530 



10.59606 
.59683 
.59760 
.59837 
.59914 




10.60766 
. 60844 
.60923 
.61001 
.61079 



10.61158 

.61236 
.61315 
.61393 
.61472 



10.61551 
.61630 
.61709 
.61788 
.61867 



10-61947 
.62026 
•62105 
•62185 
•62265 



10.62345 
.62424 
.62504 
.62585 
.62665 



10.62745 

Log.Exs. 




P. P. 



86 85 



8 


6 


8-5 


8. 


10 





9.9 


9. 


11 


4 


11.3 


11. 


12 


g 


12.7 


12. 


14 


3 


14.1 


14. 


28 


6 


28-3 


28. 


43 





42.5 


42. 


57 


3 


56.6 


56. 


71 


6 


70.8 


70. 





83 


83 


81 


6 


8.3 


8.2 


8. 


7 


9.7 


9.5 


9. 


8 


11.0 


10.9 


10. 


9 


12.4 


12.3 


12. 


10 


13.8 


13.6 


13. 


20 


27.6 


27.3 


27. 


30 


41.5 


41.0 


40. 


40 


55.3 


54.6 


54. 


50 


69.1 


68.3 


67. 





80 


79 


78 


6 


8.0 


7.9 


7. 


7 


9 


3 


9 


2 


9. 


8 


10 


6 


10 


5 


10. 


9 


12 





11 


8 


11. 


10 


13 


3 


13 


1 


13. 


20 


26 


6 


26 


3 


26. 


30 


40 





39 


5 


39. 


40 


53 




52 


6 


52. 


50 


66 


6 


65 


8 


65. 





77 


76 


7^ 


6 


7.7 


7-6 


7 


7 


9 





8 


8 


8 


8 


10 




10 


1 


10 


9 


11 


5 


11 


4 


11 


10 


12 


8 


12 


6 


12 


20 


25 


g 


25 


3 


25 


30 


38 


5 


38 





37 


40 


51 


3 


50 


6 


50 


50 


64 


1 


63 


3 


62 



84 
4 



6 

7 

8 

9 

10 

20 

30 

40 

50 



O 

0.0 
0.0 
0.0 
0.1 
0.1 
O.I 
0.2 
0.3 
0.4 



16 


^^ , 


1 6 


1 5| 


1 


8 


1 


8 


2 


1 


2 





2 


4 


2 


3 


2 


6 


2 


6 


5 


3 


5 


1 


8 





7 


7 


10 


6 


10 


3 


13 


3 


12 


9 



15 

1.5 
1-7 
2.0 
2.2 
2.5 
5.0 
7.5 
10.0 
12.5 



P. P. 



689 



TABLE VIII.— i^OGARITHMIC VERSED SINES ANB EXTERNAL SECAN^ 
80° 81° 




690 



TABLEVIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
82^" 83° 



Lg.Vers. -D Log. Exs. J> Lg. Vers 



93491 
93506 
93520 
93535 
93549 



93564 
93578 
93593 
93607 
93622 



93636 
93651 
93665 
93680 
93694 



93709 
93723 
93738 
93752 
93767 



93781 
93796 
93810 
93824 
93839 



93853 
93868 
93882 
93897 
93911 



93925 
93940 
93954 
93969 
93983 



93997 
94012 
94026 
94041 
94055 



94069 
94084 
94098 
94112 
94127 



94141 
94155 
94170 
94184 
94198 



94213 
94227 
94241 
94256 
94270 



94284 
94299 
94313 
94327 
94341 



9-94356 
Lg.Vers 



10.81806 
.81916 
.82025 
.82135 
.82245 



10.79136 
.79240 
•79345 
•79450 
.79555 



10.79660 
•79766 
•79871 
•79977 
.80083 



10.80189 
•80296 
•80402 
•80509 
•80616 



10.80723 
•80831 
.80938 
•81046 
.81154 



10.81262 
•81371 
.81479 
•81588 
•81697 




10.83470 
•83583 
•83695 
.83809 
.83922 



10.84035 
• 84149 
•84263 
•84377 
.84492 



10.84607 
•84721 
•84837 
•84952 
•85068 



10.85183 
.85299 
.85416 
.85532 
.85649 



10.85766 
Log. Exs= 



104 
105 
104 
105 
105 
105 
105 
106 
106 
106 
106 
106 
107 
107 
107 
107 
107 
108 
108 
108 
108 
108 
109 
109 
109 
109 
109 
110 
110 
110 
110 
110 
111 
111 

111 
111 
111 
112 
112 
112 
112 
112 
113 
113 

113 
114 
114 
114 
114 

115 
114 
115 
115 
116 
115 
116 
116 
116 
117 
117 
3 



9 . 94640 
.94655 
.94669 
.94683 
.94697 



9.94711 
94726 
94740 
94754 
94768 



9-94356 
•94370 
•94384 
•94398 
.94413 



. 94427 
• 94441 
•94456 
•94470 
. 94484 



. 94498 
•94512 
.94527 
.94541 
.94555 



.94569 
•94584 
•94598 
•94612 
■94626 



9.94782 
.94796 
.9481Cr 
.94825 
.94839 



9 .94853 
.94867 
.94881 
.94895 
-94909 



9 .94923 
•94938 
•94952 
.94966 
•94980 



9-94994 
•95008 
•95022 
•95036 
•95050 



9.95064 
•95078 
•95093 
•95107 
•95121 



9.95135 
•95149 
•95163 
•95177 
•95191 



9-95205 
Lg. Vers. 



Log. Exs. 



10-85766 
.85884 
.86001 
.86119 
.86237 



10.86355 
.86474 
•86592 
•86711 
.86831 



10.86950 
•87070 
•87190 
.87310 
.87431 



10-87552 
•87673 
•87794 
.87916 
•88038 



10-88160 
•88282 
•88405 
•88528 
.88651 



10.88775 
•88898 
.89022 
•89147 
•89271 



10 •89396 
.89521 
.89647 
•89773 
.89899 



10.90025 
•90152 
.90279 
•90406 
•90533 




10.91956 
.92087 
.92218 
•92350 
•92482 



10.92614 
•92747 
.92880 
.93014 
•93147 



2> 



10.93^81 
Log. Exs. 



117 
117 
117 
118 
118 
118 
118 
119 
119 

119 
120 
120 
120 
120 
121 
121 
121 
121 
122 
122 
122 
122 
123 
123 
124 
123 
124 
124 
124 
125 
125 
125 
126 
126 
126 
126 
127 
127 
127 
128 
127 
128 
129 
129 
129 
130 
129 
130 
130 

131 
131 
131 
131 
132 

132 
133 
133 
133 
133 
134 





1 
2 
3 
j4 

5 

6 

7 

8 

_9 

10 

11 

12 

13 

li 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

30 

31 
32 
33 
-34 
35 
36 
37 
38 

40 

41 
42 
43 
44 
45 
46 
47 
48 
ii 
50 
51 
52 
53 
54 
55 
56 
57 
58 
5i 
60 



P.P. 





130 


6 


13.0 


7 


15.1 


8 


17.3 


9 


19.5 


10 


21.6 


20 


43.3 


30 


65.0 


40 


86.6 


50 


108.3 





110 


6 


11.0 


7 


12.8 


8 


14.6 


9 


16.5 


10 


18.3 


20 


36.6 


30 


55.0 


40 


73.3 


50 


91^6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



3 

0.3 
0.3 
0.4 
C.4 
0.5 
1.0 
1.5 
2.0 
2.5 





1 


6 


0.11 


7 







8 







9 







10 







20 





3 


30 





5 


40 





6 


50 





8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



13 

1.4 
1.7 
1.9 
2.2 
2.4 
4.8 
7.2 
9.6 
12.1 



120 

12.0 
14.0 
16.0 
18.0 
20.0 
40.0 
60.0 
80.0 
100.0 



100 

10^0 
11.6 
13.8 
15.0 
16-6 
33.3 
50.0 
66.6 
83.3 



2 

0^2 
0^2 
0^2 
0.3 
0-3 
0.6 
1^0 
1.3 
1.6 



O 

o.i 

0.0 
0.0 
0.1 
0.1 
O.I 
0.2 
0.3 
0.4 



14 

1.4 
1^6 
1^8 
2^1 
2-3 
4-6 
7^0 
9.3 
11.6 



P.P. 



e9i 



TABLE VIII.— logahithmic versed sines and external secants. 

84* 85° 



Lg.Vers. I> Log.Exs. J> Lg.Vers 



O 

1 
2 
3 

5 

6 

7 

8 

_9 

10 

11 

12 

13 

li 

15 

16 

17 

18 

19 



9-95205 
.95219 
.95233 
.95247 
•95261 



20 

21 
22 
23 
24 

25 
26 
27 
28 
29 



9.95275 
.95289 
.95303 
.95317 
.95331 



•95345 
.95359 
.95373 
.95387 
•95401 



•95415 
•95429 
.95443 
• 95457 
•95471 



•95485 
•95499 
•95513 
• 95527 
•95540 



•95554 
•95568 
•95582 
•95596 
•95610 



30 

31 
32 
33 
34 



35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
46 
47 
48 
49 



50 

51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



•95624 
•95638 
•95652 
•95666 
95680 



•95693 
•95707 
.95721 
.95735 
•95749 



•95763 
•95777 
.95791 
•95804 
•95818 



•95832 
.95846 
.95860 
.95874 
•95888 



9.95901 
.95915 
.95929 
.95943 
•95957 



9-95970 
95984 
-95998 
•96012 
•96026 



9 •96039 
Lg. Vers. 



14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
13 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
14 
13 

14 
14 
14 
14 
14 
14 
13 
14 
14 
14 
13 
14 
14 
14 
14 
13 
14 
14 
13 
14 
14 
14 
13 
14 
14 

13 
14 
13 
14 
14 

13 
14 
14 
13 
14 
13 



10. 



93281 
93416 
93551 
93686 
93821 



10. 



93957 
94093 
94229 
94366 
94503 



10. 



94641 
94778 
94917 
95055 
95194 



10 



95333 
95473 
95613 
95753 
95894 



10. 



96035 
96176 
96318 
96461 
96603 



10. 



96746 
96889 
97033 
97177 
97322 



10 



.97467 
.97612 
.97758 
.97904 
•98050 



10. 



98197 
98345 
98492 
93640 
98789 



10.98938 
.99087 
.99237 
.99387 
-99538 



10 



•99689 
-99841 
-99993 
-00145 
-00298 



11. 



00451 
00605 
00759 
00914 
01069 



11 



•01225 
-01381 
.01537 
.01694 
-01852 



11-02010 
Log.Exs, 



134 
135 
135 
135 
135 
136 
136 
137 
137 
137 
137 
138 
138 
139 
139 
139 
140 
140 
140 
14l 
141 
142 
142 
142 
143 
143 
144 
144 
144 
145 
145 
145 
146 
146 
147 
147 
147 
148 
149 
149 
149 
150 
150 
151 

151 
151 
152 
152 
153 
153 
154 
154 
155 
155 
155 
156 
156 
157 
157 
158 



-96039 
.96053 
.96067 
.96081 
-96095 



96108 
96122 
96136 
96150 
96163 



96177 
96191 
.96205 
.96218 
.96232 



.96246 
.96259 
.96273 
.96287 
-96301 



-96314 
.96328 
.96342 
.96355 
-96369 



-96383 
•96397 
-96410 
-96424 
-96438 



-96451 
.96465 
.96479 
.96492 
•96506 



-96519 
-96533 
-96547 
-96560 
-96574 



•96588 
-96601 
.96615 
.96629 
•96642 



.96656 
.96669 
.96683 
•96697 
•96710 



•96724 
•96737 
-96751 
-96764 
•96778 



•96792 
•96805 
-96819 
-96832 
-96846 



9-96859 



D Lg.Vers 



11 



Log.Exs. I Xi» 



02010 
02168 
02327 
02487 
02646 



11. 



02807 
02968 
03129 
03291 
03453 



11. 



03616 
03780 
03944 
04108 
04273 



11. 



04438 
04604 
04771 
04938 
05106 



11 



.05274 
.05443 
.05612 
.05782 
•05952 



11. 



06123 
06295 
06467 
06640 
06813 



11. 



06987 
07161 
07336 
07512 
07688 



11. 



07865 
08043 
08221 
08400 
08579 



11 



08759 
08940 
09121 
09303 
09486 



11 



09669 
.09853 
.10038 
■10223 

10409 



11 



10595 
10783 
10971 
11160 
11349 



11 



11539 
11730 
11922 
12114 
12307 



1112501 

T> Log.Exs. 



158 
159 
159 
159 
160 
161 
161 
161 
162 
163 
163 
164 
164 
165 
165 
166 
167 
167 
167 
168 
169 
169 
169 
170 

171 
171 
172 
173 
173 
174 
174 
175 
176 
176 

177 
177 
178 
179 
179 
180 
180 
181 
182 
182 

183 

184 
185 
185 
186 
186 
187 
188 
189 
189 
190 
191 
191 
192 
193 
193 



O 

1 
2 
3 
_4 
5 
6 
7 
8 

10 

11 
12 
13 
1£ 

15 
16 
17 
18 

20 

21 
22 
23 
24 

25 
26 
27 
28 

30 

31 
32 
33 
M. 
35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

45 
46 
47 
48 
j49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
_51 
60 



P.P. 





190 


6 


19-0 


7 


22-1 


8 


25-3 


9 


28-5 


10 


31-6 


20 


63-3 


30 


95-0 


40 


126-6 


50 


158.3 



170 



9 
10 
20 
30 
40 
50 



10 
20 
30 
40 
50 



17 





16- 


19 


8 


18- 


22 


6 


21- 


25 


5 


24- 


28 


3 


26- 


56 


6 


53- 


85 





80- 


113 


3 


106- 


141 


6 


133 



150 



15 





14- 


17 


5 


16. 


20 





18 • 


22 


5 


21. 


25 





23- 


50 





46- 


75 





70 • 


100 





93 • 


125 





116^ 



180 

18.0 
21-0 
24.0 
27.0 
30.0 
60.0 
90.0 
120.0 
150.0 

160 


6 
8 

6 
3 




140 


3 
6 

3 



13 








9 


0. 


15 


1 


1 





0^ 


17 


3 


1 


z 


1^ 


19 


5 


1 


3 


1. 


21 


6 


1 


5 


1^ 


43 


3 


3 





2- 


65 





4 


5 


4^ 


86 


6 


6 





5^ 


108 


3 


7 


5 


6. 



130 9 8 

• 8 

• 9 

• 
■2 
•3 

• 6 

• 
•3 

6 



5 

0-5 
0-6 
0-6 
0-7 
0^8 
1-6 
2^5 
3-3 
4-1 





7 


( 


^ 


6 


0-7 


0-6 


7 


0^8 





7 


8 


0^9 





8 


9 


1^0 





9 


10 


1^1 


1 





20 


2^3 


2 





30 


3^5 


3 





40 


4^6 


4 





50 


5-8 


5 






14 

1-4 
1-7 
1-9 
2-2 
2^4 
4-8 
7-2 
9^6 
12-1 



14 

1-4 
1-6 
1-8 
2.] 
2.3 
4^6 
7.0 
93 
11^6 



13 

1-3 



P. P. 



692 



' TABLE YIII.— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 

86° 87° 



Lg. Vers. 



96859 
96837 
96887 
96900 
96914 



96927 
96941 
96954 
96968 
96981 



96995 
97008 
97022 
97035 
97049 



97062 
97076 
97089 
97103 
97116 



97130 
97143 
97157 
97170 
97183 



97197 
97210 
97224 
97237 
97251 



97264 
97277 
97291 
97304 
97318 
97331 
97345 
97358 
97371 
97385 



97398 
97412 
97425 
97438 
97452 



97465 
97478 
97492 
97505 
97519 



97532 
97545 
97559 
97572 
97585 



97599 
97612 
97625 
97639 
97652 



9.97665 
Lg.Vers, 



D 



Log. Exs, 



11.12501 
.12696 
•12891 
.13087 
.13284 



11.13482 
•13680 
.13879 
.14079 
.14280 



11.14482 
•14684 
.14887 
.15092 
•15297 



11^15502 
.15709 
.15917 
.16125 
.16334 



11.16544 
.16755 
.16967 
.17180 
.17394 



11.17609 
.17824 
.18041 
.18259 
•18477 



11^18697 
.18917 
.19138 
.19361 
•19584 



11.19809 
.20034 
.20261 
.20489 
•20717 



11^20947 
.21178 
.21410 
.21643 
•21877 



11-22112 
.22349 
.22586 
.22825 
.23065 



11.23306 
•23548 
•23792 
•24037 
•24283 



11^24530 
.24778 
.25028 
.25279 
.25531 



11.25785 
Log. Exs, 



2> 



195 
195 
196 
196 
198 
198 
199 
200 
201 
201 
202 
203 
204 
205 
205 
206 
208 
208 
209 
210 
211 
212 
213 
214 
214 
215 
216 
218 
218 
219 
220 
221 
222 
223 
224 
225 
227 
227 
228 
230 
230 
232 
233 
234 

235 

236 
237 
239 
239 
24l 
242 
243 
245 
246 
247 
248 
250 
251 
252 
254 



Lg.Vers. 




97665 
97679 
97692 
9770_ 
97718 



9786 

97878 

97891 

97904 

97917 



97931 
97944 
97957 
97970 
97984 



97997 
98010 
98023 
98036 
98050 



98063 
98076 
98089 
98102 
98116 



98129 
98142 
98155 
98168 
98181 




98326 
98339 
98352 
98365 
98378 



98392 
98405 
98418 
98431 
98444 



9-98457 
Lg, Vers, 



2> 



Log, Exs, 



il. 25785 
.26040 
.26297 
.26554 
.26814 



11.27074 
.27336 
.27599 
.27864 
.28131 



11.28398 
.28668 
.28938 
.29211 
.29485 



11.29760 
.30037 
.30316 
.30596 
.30878 



11.31162 
.31447 
.31734 
.32023 
.32313 



11.32606 
.32900 
.33196 
.33494 
.33793 



11.34095 
.34398 
.34704 
•85011 
•35321 



11.35632 
.35946 
•36261 
•36579 
•36899 



11.37221 
.37546 
.37872 
.38201 
.38532 



11.38866 
.39201 
•39540 
•39880 
•40224 



11.40569 
•40918 
.41269 
.41622 
.41979 



11.42338 
.42699 
•43064 
•43431 
•43802 



11-44175 
Log, Exs. 



2> 



255 
256 
257 
259 
260 
262 
263 
265 
266 
267 
269 
270 
272 
274 
275 
277 
278 
280 
282 
283 
285 
287 
288 
290 
292 
294 
296 
298 
299 
301 
303 
305 
307 
309 
311 
313 
315 
318 
320 
322 
324 
326 
328 
331 
333 
335 
338 
340 
343 
345 
348 
351 
353 
356 
359 
361 
364 
367 
370 
373 



P. P. 





250 


6 


25.01 


7 


29 


1 


8 


33 


3 


9 


37 


5 


10 


41 


6 


20 


83 


3 


30 


125 





40 


166 


6 


50 


208 


3 





330 


6 


23^0 


7 


26-8 


8 


30.6 


9 


34.5 


10 


38-3 


20 


76.6 


30 


115-0 


40 


153.3 


50 


191.6 



6 

7 

8 

9 

10 

20 

30 

40 

50 



310 

21.0 

24.5 

28.0 

31.5 

35.0 

70.0 

105.0 

140.0 

175.0 

190 

19.0 
22.1 
25-3 
28.5 
31.6 
63-8 
95.0 
126.6 
158-3 



340 

24.0 

28.0 

32.0 

36-0 

40.0 

80.0 

120-0 

160.0 

200.0 

330 

22.0 

25.6 

29.3 

33.0 

36.6 

73.3 

110.0 

146.6 

183.3 

300 

20.0 

23.3 

26.6 

30.0 

33.3 

66.6 

100.0 

133.3 

166.6 



4 

0.4 
0.4 
0.5 
0.6 
0.6 
1^3 
2^0 
2^6 
3^3 



3 

0.3 
0-3 
0.4 
0.4 
0-5 
1.0 
1.5 
2.0 
2.5 





3 


1 


6 


0.2 


0.1 


7 


0.2 


0.1 


8 


0.2 


0-1 


9 


0-3 


0-1 


10 


0.3 


0.1 


20 


0^6 


0.3 


30 


1.0 


0.5 


40 


1.3 


0.6 


50 


1.6 


0.8 





14 


13 


6 


1.4 


1.3 


7 


1.6 


1.6 


8 


1.8 


1^8 


9 


2.1 


2^0 


10 


2-3 


2^2 


20 


4.6 


4^5 


30 


7.0 


6-7 


40 


9.3 


9-0 


50 


11-6 


11-2 






P.P. 



0% 

0^0 
0-0 
0.1 
0^1 
0^1 
0^2 
0^3 
0.4 

13 

1.3 
1.5 
1.7 
1.9 
2.1 
4.3 
6^5 
8-6 
10.8 



693 



TABLE VIIl— LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS. 
88^* 89° 



Lg. Vers, 



98457 
98470 
98483 
98496 
98509 



98522 
98535 
98548 
98562 
98575 



98588 
98601 
98614 
98627 
98640 



98653 
98666 
98679 
98692 
98705 



98718 
98731 
98744 
98757 
98770 



98783 
98796 
98809 
98822 
98835 



98848 
98861 
98874 
98887 
98900 



98913 
98925 
98938 
98951 
98964 



98977 
98990 
99003 
99016 
99029 



99042 
99055 
99068 
99081 
99093 



99106 
99119 
99132 
99145 
99158 



99171 
99184 
99197 
99209 
99222 
99235 
Lg. Vers. 



Z> 



Log.Exs. 




11.48083 
•48493 
.48906 
.49323 
.49743 



11.50168 
.50597 
.51029 
.51466 
.51906 



11.52351 
.52801 
.53255 
.53713 
•54176 



D 



11.54643 
.55116 
.55593 
.56076 
.56563 



11.57056 
.57554 
.58058 
.58567 
.59082 



11.59602 
.60129 
.60662 
.61202 
•61747 



11.62300 
.62859 
.63425 
.63998 
•64579 



11.65167 
.65762 
.66366 
.66978 
■67598 



11.68227 
.68865 
.69511 
.70168 
.70834 



11.71509 
.72196 
.72892 
.73600 
•74319 



11 75050 
Log.Exs. 



376 
379 
382 
386 
389 
392 
395 
399 
402 
406 
409 
413 
417 
420 
425 
428 
432 
436 
440 
445 
449 
454 
458 
463 
467 
472 
477 
482 
487 
492 
498 
504 
509 
515 
520 
527 
533 
539 
545 
552 
559 
566 
573 
581 
588 
595 
604 
611 
620 
628 
638 
646 
656 
666 

675 
686 
696 
707 
719 
73(5 

i7 



Lg. Vers, 



•99235 
.99248 
.99261 
.99274 
.99287 



.99299 
.99312 
.99325 
.99338 
.99351 



.99363 
.99376 
.99389 
.99402 
.99415 



.99428 
.99440 
.99453 
.99466 
.99479 



2> 



.99491 
.99504 
.99517 
.99530 
•99543 



•99555 
.99568 
.99581 
.99594 



•99619 
.99632 
.99645 
.99657 
.99670 



•99683 
.99695 
.99708 
.99721 
■99734 



•99746 
.99759 
.99772 
.99784 
•99797 



•99810 
.99823 
.99835 
.99848 
.99861 



.99873 
.99886 
.99899 
.99911 
.99924 



.99937 
.99949 
.99962 
.99974 
•99987 



Log.Exs, 



11 •75050 
•75792 
.76547 
.77316 
.78097 



11.78892 
.79702 
.80527 
.81367 

.8 222 3 

11 •83095 
.83986 
.84894 
.85821 
.86768 



11.87735 
.88724 
.89735 
.90769 
•91829 



J> 




12^05535 
.07020 
.08557 
.10149 
•11801 



12^13517 
.15302 
.17163 
•19106 
■21139 




12^53501 
•58089 
.63217 
.69029 
■75736 




Log.Exs, 



D 



742 
755 
768 
781 
795 
809 
825 
840 
856 
872 
890 
908 
927 
947 
967 
989 
1009 
1034 
1059 
1085 
1112 
1140 
1171 
1203 
1236 
1271 
1309 
1349 
1391 
1436 
1485 
1537 
1592 
1652 
1716 
1785 
1861 
1943 
2033 
2131 
2240 
2361 
2495 
2645 
2815 
3009 
3231 
3489 
3791 
4152 
4588 
5127 
5812 
6707 
7931 
9704 
12506 
17621 
30116 



2> 





1 

2 

3 

4 

5 

6 

7 

8 

_9 

10 

11 

12 

13 

15 
16 
17 
18 
11 
30 
21 
22 
23 
24 
25 
26 
27 
28 
29. 
30 
31 
32 
33 
-34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
lii 
45 
46 
47 
48 
_49^ 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 



I 



P. P, 





13 


6 


1.3 


7 


1.6 


8 


1.8 


9 


2.0 


10 


2.2 


20 


4.5 


30 


6.7 


40 


9.0 


50 


11.2 



13 

1.3 
1.5 
!•? 

\i 

4.3 

6.5 

8.6 

10.8 



6 

7 

8 

9 

10 

20 

30 

40 

50 



13 

1.2 
1.4 
1.6 
1-9 
21 
4^1 
6.2 
8^3 
10.4 



P. P. 



694 



? / TABLE IX.->NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 





Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


# 




1 

2 
3 

4 


.00000 
.00029 
.00058 
.00087 
.00116 


One 
One 
One 

One 
One 


.00000 
.00029 
.00058 
.00087 
.00116 


Infinite 
3437.75 
1718.87 
1145.92 
859.436 


.01745 
.01774 
.01803 
.01832 
.01862 


.99985 
.99984 
.99984 
•99983 
.99983 


.01746 
.01775 
.01804 
.01833 
.01862 


57.2900 
56.3506 
55.4415 
54.5613 
53-7086 


60 

59 
58 
57 
58 


5 
6 
7 
8 
9 


.00145 
.00175 
.00204 
.00233 
.00262 


One 
One 
One 
One 
One 


.00145 
.00175 
.00204 
.00233 
.00262 


687.549 
572.957 
491.106 
429.718 
381.971 


.01891 
.01920 
.01949 
.01978 
.02007 


•99982 
•99982 
.99981 
.99980 
.99980 


.01891 
.01920 
.01949 
.01978 
.02007 


52.8821 
52.0807 
51.3032 
50.5485 
49.8157 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.00291 
.00320 
.00349 
.00378 
.00407 


One 
.99999 
.99999 
.99999 
.99999 


.00291 
.00320 
.00349 
.00378 
.00407 


343.774 
312.521 
286.478 
264.441 
245.552 


.02036 
.02065 
.02094 
.02123 
.02152 


.99979 
.99979 
.99978 
.99977 
.99977 


.02036 
.02066 
.02095 
.02124 
.02153 


49.1039 
48.4121 
47.7395 
47.0853 
46.4489 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.00436 

.00465 
.00495 
.00524 
.00553 


.99999 
.99999 
.99999 
.99999 
.99998, 


.00436 
.00465 
.00495 
.00524 
.00553 


229.182 
214-858 

202.219 
190.984 
180.932 


.02181 
.02211 
.02240 
.02269 
.02298 


.99976 
.99976 
.99975 
.99974 
•99974 


.02182 
.02211 
.02240 
.02269 
•02298 


45.8294 
45.2261 
44.6386 
44.0661 
43.5081 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.00582 
.00611 
.00640 
.00669 
.00698 


.99998 
.99998 
.99998 
.99998 
.99998 


.00582 
.00611 
.00640 
.00669 
.00698 


171.885 
163.700 
156.259 
149.465 
143.237 


.02327 
.02356 
.02385 
.02414 
.02443 


.99973 
.99972 
.99972 
.99971 
.99970 


.02328 
.02357 
.02386 
.02415 
.02444 


42.9641 
42.4335 
41.9158 
41.4106 
40.9174 


40 

39 
38 
37 

36 


25 
26 
27 
28 
29 


.00727 
.00756 
Z)0785 
.00814 
.00844 


.99997 
.99997 
.99997 
.9C997 
.99996 


.00727 
.00756 
.00785 
.00813 
.00844 


137.507 
132.219 
127.321 
122.774 
118.540 


.02472 
.02501 
.02530 
.02560 
.02589 


.99969 
.99969 
.99968 
.99967 
.99966 


.02473 
.02502 
.02531 
.02560 
.02589 


40.4358 
39.9655 
39.5059 
39.0568 
38.6177 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.00873 
.00902 
.00931 
.00960 
.00989 


.99998 
.99998 
.99996 
.99995 
.99995 


.00873 
.00902 
.00931 
.00960 
.00989 


114.589 
110.892 
107.426 
104.171 
101.107 


.02618 
.02647 
.02676 
.02705 
.02734 


.99966 
.99965 
.99964 
.99963 
.99963 


.02619 
.02648 
.02677 
.02706 
•02735 


38.1885 
37.7686 
37.3579 
36.9560 
36.5627 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.01018 
.01047 
.01076 
.01105 
.01134 


.99995 
.99995 
.99994 
.99994 
.99994 


.01018 
.01047 
.01076 
.01105 
.01135 


98.2179 
95.4895 
92.9085 
90.4633 
88.1436 


.02763 
.02792 
.02821 
.02850 
.02879 


.99962 
.99961 
.99960 
.99959 
•99959 


•02764 
.02793 
.02822 
.02851 
.02881 


36.1776 
35-8006 
35.4313 < 
35.0695 
34.7151 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.01164 

.01193 

.01222. 

.01251 

.01280 


.99993 
.99993 
.99993 
.99992 
.99992 


.01164 
.01193 
.01222 
.01251 
.01280 


85-9398 
83.8435 
81.8470 
79.9434 
78.1263 


.02908 
.02938 
.02967 
.02996 
.03025 


.99958 
•99957 
.99956 
.99955 
.99954 


.02910 
.02939 
.02968 
.02997 
.03026 


34.3678 
34.0273 
33.6935 
33.3662 
33.0452 


30 

19 
18 
17 
16 


46 
46 
47 
48 
49 


.01309 
.01338 
.01367 
.01396 
.01425 


.99991 
.99991 
.99991 
.99990 
.99990 


.01309 
.01338 
.01367 
.01396 

.01425 


76.3900 
74.7292 
73.1390 
71.6151 
70.1533 


•03054 
.03083 
•03112 
•03141 
•03170 


.99953 
.99952 
.99952 
.99951 
.99950 


.03055 
.03084 
.03114 
.03143 
.03172 


32.7303 
32.4213 
32.1181 
31.8205 
31-5234 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.01454 
.01483 
.01513 
.01542 
.01571 


.99989 
.99989 
.99989 
.99988 
.99988 


.01455 
.01484 
.01513 
.01542 
.01571 


68.7501 
67.4019 
66.1055 
64.8580 
63.6567 


.03199 
•03228 
.03257 
.03286 
.03316 


.99949 
.99948 
.99947 
.99946 
.99945 ' 


.03201 
.03230 
•03259 
.03288 
.03317 


31.2416 
30.9599 
30.6833 
30.4116 
30.1446 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.01600 
.01629 
.01658 
.01687 
.01716 


.99987 
.99987 
.99986 
.99986 
.99985 


.01600 
.01629 
.01658 
.01687 
.01716 


62.4992 
61.3829 
60.3058 
59.2659 
58.2612 
57.2900 


.03345 
.03374 
•03403 
•03432 
.03461 


.99944 
.99943 
.99942 
.99941 
.99940 


.03346 
.03376 
.03405 
•03434 
.03463 


29.8823 
29.6245 
.Q9.3711 
29.1220 
28-8771 


5 

4 
3 

2 

1 


60 


.01745 


.99985 


.01746 


.03490 


.99939 


.03492 


28-6363 
Tan. 





' 


Cos. 1 Sin. 


Cot. 1 Tan. | 


Cos. 


Sin. Cot. 


/ 



89** 



695 



88° 



TABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
3° 3^ 


/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 1 Cot. ^ 


/ 



1 
2 
3 

4 


.03490 
.03519 
.03548 
.03577 
.03606 


.99939 
.99938 
.99937 
.99936 
.99935 


.03492 
.03521 
.03550 
.03579 
.03609 


28.6363 
28.3994 
28.1664 
27-9372 
27.7117 


.05234 
.05263 
.05292 
.05321 
.05350 


.99863 
.99861 
.99860 
.99858 
.99857 


.05241 
.05270 
.05299 
.05328 
-05357 


19.0511 
18-^55 
18-8711 
18-7678 
18-6656 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 


5 
6 
7 
8 
9 


.03635 
.03664 
.03693 
.03723 
.03752 


.99934 
.99933 
.99932 
.99931 
.99930 


.03638 
.03667 
.03696 
.03725 
.03754 


27.4899 
27.2715 
27.0566 
26.8450 
26.6367 


-05379 
.05408 
.05437 
.05466 
.05495 


.99855 
.99854 
.99852 
.99851 
.99849 


.05387 
.05416 
.05445 
.05474 
.05503 


18-5645 
18-4645 
18-3655 
18.2677 
18-1708 


10 

11 
12 
13 
14 


.03781 
.03810 
.03839 
.03868 
.03897 


.99929 
.99927 
.99926 
.99925 
.99924 


.03783 
.03812 
.03842 
.03871 
.03900 


26.4316 
26.2296 
26.0307 
25.8348 
25.6418 


.05524 
.05553 
.05582 
-05611 
.05640 


.99847 
.99846 
.99844 
.99842 
.99841 


.05533 
.05562 
.05591 
.05620 
.05649 


18-0750 
17.9802 
17.8863 
17.7934 
17-7015 


15 
16 
17 
18 
19 


.03926 
.03955 
.03984 
.04013 
.04042 


.99923 
.99922 
.99921 
.99919 
.99918 


.03929 
.03958 
.03987 
.04016 
.04046 


25.4517 
25.2644 
25-0798 
24.8978 
24.7185 


.05669 
-05698 
.05727 
.05756 
.05785 


.99839 
.99838 
.99836 
.99834 
.99833 


-05678 
.05708 
.05737 
.05766 
.05795 


17.6106 
17.5205 
17.4314 
17-3432 
17-2558 


45 
44 
43 
42 
4i 


30 

21 
22 
23 
24 


.04071 
.04100 
.04129 
.04159 
.04188 


.99917 
.99916 
.99915 
.99913 
.99912 
.99911 
.99910 
.99909 
.99907 
.99906 


.04075 
.04104 
.04133 
.04162 
.04191 


24.5418 
24-3675 
24-1957 
24-0263 
23.8593 


.05814 
.05844 
-05873 
-05902 
.05931 


.99831 
.99829 
-99827 
-99826 
.99824 


.05824 
.05854 
.05883 
-05912 
.05941 


17-1693 
17.0837 
16.9990 
16-9150 
16.8319 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.04217 
.04246 
.04275 
.04304 
.04333 


.04220 
.04250 
.04279 
.04308 
.04337 
.04366 
.04395 
.04424 
.04454 
.04483 


23-6945 
23.5321 
23-3718 
23-2137 
23.0577 


.05960 
.05989 
.06018 
.06047 
.06076 


-99822 
-99821 
-99819 
.99817 
.99815 


-05970 
-05999 
-06029 
-06058 
-06087 


16-7496 
16-6681 
16.5874 
16.5075 
16.4283 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.04362 
.04391 
.04420 
.04449 
.04478 


.99905 
.99904 
•99902 
.99901 
.99900 


22-9038 
22.7519 
22-6020 
22-4541 
22.3081 


.06105 
.06134 
.06163 
.06192 
.06221 


.99813 
.99812 
.99810 
.99808 
.99806 


.06116 
.06145 
-06175 
.06204 
.06233 


16.3499 
16-2722 
16-1952 
16-1190 
16.0435 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.04507 
.04536 
.04565 
.04594 
.04623 


.99898 
.99897 
.99896 
.99894 
.99893 


.04512 
.04541 
.04570 
.04599 
.04628 


22-1640 
22.0217 
21-8813 
21-7426 
21.6056 


.06250 
.06279 
.06308 
.06337 
.06366 


.99804 
.99803 
.99801 
.99799 
.99797 


.06262 
.06291 
.06321 
.06350 
.06379 


15.9687 
15.8945 
15.8211 
15.7483 
15.6762 


25 

24 
23 
22 
21 


40 

41 
42 
43 
44 


.04653 
.04682 
.04711 
.04740 
.04769 


.99892 
.99890 
.99889 
.99888 
.99886 


.04658 
.04687 
.04716 
.04745 
.04774 


21.4704 
21.3369 
21-2049 
21.0747 
20.9460 


.06395 
.06424 
.06453 
.06482 
.06511 


.99795 
.99793 
.99792 
.99790 
.99788 


.06408 
.06437 
.06467 
.06496 
.06525 


15.6048 
15.5340 
15.4638 
15.3943 
15.3254 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.04798 
.04827 
.04856 
.04885 
.04914 


.99885 
.99883 
.99882 
.99881 
.99879 


.04803 
.04833 
■04862 
.04891 
.04920 


20.8188 
20.6932 
20.5691 
20.4465 
20.3253 


.06540 
.06569 
.06598 
.06627 
.06656 


.99786 
.99784 
.99782 
.99780 
.99778 


.06554 
.06584 
.06613 
.06642 
.06671 


15.2571 
15.1893 
15.1222 
15.0557 
14.9898 
14.9244 
14.8596 
14.7954 
14.7317 
14.6685 


15 

14 

13 

12 

11 

10 

9 

8 

7 

6 


60 

51 
32 
53 
54 


.04943 
.04972 
.05001 
.05030 
.05059 
.05088 
.05117 
.05146 
.05175 
.05205 


.99878 
.99876 
.99875 
.99873 
.99872 


.04949 
.04978 
.05007 
.05037 
.05066 


20.2056 
20.0872 
19.9702 
19.8546 
19.7403 


.06685 
.06714 
.06743 
.06773 
.06802 


.99776 
.99774 
.99772 
.99770 
.99768 


.06700 
.06730 
-06759 
.06788 
.06817 


55 
56 
57 

h9 


.99870 
.99869 
.99867 
.99866 
.99864 
.99863 


.05095 
.05124 
.05153 
.05182 
.05212 


19.6273 
19.5156 
19.4051 
19.2959 
19.1879 
19.0811 


.06831 
.06860 
.06889 
.06918 
.06947 
.06976 


.99766 
.99764 
.99762 
.99760 
.99758 


.06847 
-06876 
.06905 
.06934 
.06963 


14.6059 
14.5438 
14.4823 
14.4212 
14.3607 
14.3007 


5 

4 
3 
2 
1 

d 


60 


.05234 


.05241 


.99756 


.06993 


/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. ' 



67'' 



696 



86° 



! TABLE IX.- 


-NATURAL SINES, COSINES. TANGENTS. AND COTANGENTa 
4° 5° 


9 


Sin. 


Cos. 


TaD. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 

4 


.06976 
.07005 
.07034 
•07063 
.07092 


.99756 
.99754 
.99752 
.99750 
.99748 


.06993 
.07022 
.07051 
.07080 
.07110 


14.3007 
14.2411 
14.1821 
14.1235 
14.0655 


.08716 
.08745 
.08774 
.08803 
.08831 


.99619 
.99617 
.99614 
.99612 
.99609 


.08749 
.08778 
.08807 
.08837 
.08866 


11.4301 
11.3919 
11.3540 
11.3163 
11.2789 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.07121 
.07150 
.07179 
.07208 
.07237 


.99746 
.99744 
.99742 
.99740 
.99738 


.07139 
.07168 
.07197 
.07227 
.07256 


14.0079 
13.9507 
13.8940 
13.8378 
13.7821 


.08860 
.08889 
.08918 
.08947 
.08976 


•99607 
•99604 
.99602 
.99599 
.99596 


.08895 
•08925 
•08954 
•08983 
.09013 


11.2417 
11.2048 
11.1681 
11.1316 
11.0954 


55 
54 
53 
52 
51 


10 

11 

12 
13 

14 


.07266 
.07295 
.07324 
.07353 
.07382 


.99736 
.99734 
.99731 
.99729 
•99727 


.07285 
.07314 
.07344 
.07373 
.07402 


13.7267 
13.6719 
13.6174 
13.5634 
13.5098 


.09005 
.09034 
.09063 
.09092 
.09121 


.99594 
.99591 
.99588 
.99586 
.99583 


•09042 
.09071 
.09101 
.09130 
.09159 


11.0594 
11.0237 
10.9882 
10.9529 
10.9178 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.07411 
.07440 
.07469 
.07498 
.07527 


.99725 
.99723 
.99721 
.99719 
.99716 


.07431 
.07461 
.07490 
.07519 
.07548 


13.4566 
13.4039 
13.3515 
13.2996 
13.2480 


.09150 
.09179 
.09208 
.09237 
.09266 


.99580 
.99578 
.99575 
.99572 
.99570 


.09189 
.09218 
.09247 
.09277 
.09306 


10.8829 
10.8483 
10.8139 
10.7797 
10.7457 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


.07556 
.07585 
.07614 
.07643 
.07672 


.99714 
.99712 
.99710 
.99708 
.99705 


.07578 
.07607 
.07636 
.07665 
.07695 


13.1969 
13.1461 
13.0958 
13.0458 
12.9962 


.09295 
.09324 
.09353 
.09382 
.09411 


.99567 
.99564 
.99562 
.99559 
.99556 


.09335 
.09365 
.09394 
.09423 
.09453 


10.7119 
10.6783 
10.6450 
10.6118 
10.5789 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.07701 
.07730 
.07759 
.07788 
.07817 


.99703 
.99701 
.99699 
.99696 
.99694 


.07724 
.07753 
.07782 
.07812 
.07841 


12.9469 
12.8981 
12.8496 
12.8014 
12.7536 


.09440 
.09469 
.09498 
.09527 
.09556 


.99553 
.99551 
.99548 
.99545 
.99542 


.09482 
.09511 
.09541 
.09570 
.09600 


10.5462 
10.5136 
10.4813 
10.4491 
10.4172 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.07846 
.07875 
.07904 
.07933 
.07962 


.99692 
.99689 
.99687 
.99685 
.99683 


.07870 
.07899 
.07929 
.07958 
.07987 


12.7062 
12.6591 
12.6124 
12.5660 
12.5159 


.09585 
.09614 
.09642 
.09671 
.09700 


.99540 
.99537 
.99534 
.99531 
.99528 


.09629 
.09658 
.09688 
.09717 
.09746 


10.3854 
10.3538 
10.3224 
10.2913 
10.2602 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


•07991 
.08020 
.08049 
.08078 
.08107 


.99680 
.99678 
.99676 
.99673 
.99671 


.08017 
.08046 
.08075 
.08104 
.08134 


12.4742 
12.4288 
12.3838 
12.3390 
12.2946 


•09729 
.09758 
.09787 
•09816 
.09845 


.99526 
•99523 
.99520 
•99517 
99514 


.09776 
.09805 
•09834 
•09864 
.09893 


10.2294 
10.1988 
10.1683 
:.0.1381 
10.1080 


25 
24 
23 
22 
.21 


40 

41 
42 
43 
44 


.08136 
.08165 
.08194 
.08223 
.08252 


.99668 
.99666 
.99664 
.99661 
.99659 


.08163 
.08192 
.08221 
.08251 
.08280 


12.2505 
12.2067 
12.1632 
12.1201 
12.0772 


•09874 
.09903 
•09932 
.09961 
.09990 


.99511 
.99508 
.99506 
.99503 
.99500 


.09923 
.09952 
.09981 
.10011 
.10040 


10.0780 
10.0483 
10. 0187 
9.98931 
9.96007 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.08281 
.08310 
.08339 
.08368 
.08397 


.99657 
.99654 
.99652 
.99649 
.99647 


.08309 
.08339 
.08368 
.08397 
.08427 


12.0346 
11.9923 
11.9504 
11.9087 
11.8673 


.10019 
.10048 
.10077 
.10106 
.10135 


.99497 
.99494 
.99491 
.99488 
.99485 


.10069 
.10099 
.10128 
.10158 
.10187 


9.93101 
9.90211 
9.87338 
9.84482 
9.81641 


15 
14 
13 
12 
-11 


50 

51 
52 
53 

54 


.08426 
.08455 
.08484 
.08513 
.08542 


.99644 
.99642 
.99639 
.99637 
.99635 


.08456 
.08485 
.08514 
.08544 
.08573 


11.8262 
11.7853 
11.7448 
11.7045 
11.6645 


.10164 
.10192 
•10221 
.10250 
•10279 


.99482 
.99479 
.99476 
.99473 
.99470 


.10216 
.10246 
.10275 
.10305 
.10334 


9.78817 
9.76009 
9.73217 
9.70441 
9.67680 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.08571 
.08600 
.08629 
.08658 
.08687 


.99632 
.99630 
.99627 
.99625 
.99622 


.08602 
.08632 
.08661 
.08690 
.08720 


11.6248 
11.5853 
11.5461 
11.5072 
11.4685 


•10308 
•10337 
.10366 
.10395 

.10424 


.99467 
.99464 
.99461 
.99458 
.99455 


.10363 
.10393 
.10422 
.10452 
.10481 


9.64935 
9.62205 
9.59490 
9.56791 
9.54106 


5 
4 
3 
2 
1 


60 


08716 


.99619 


.08749 


11.4301 


•10453 


.99452 


.10510 


9.51436 







Cos. 


Sin. 


Cot. 


Tan. 


Cos. Sin. I Cot. 


Tan. 





85^ 



697 



84* 



1 ABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. P 







6 


o 






^o 






1 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. ' 




1 

2 
3 

4 


.10453 
.10482 
.10511 
.10540 
.10569 


.99452 
.99449 
.99446 
.99443 
.99440 


.10510 
.10540 
.10569 
.10599 
.10628 


9.51436 
9.48781 
9.46141 
9.43515 
9.40904 


12187 
.12216 
.12245 
.12274 
.12302 


.99255 
.99251 
.99248 
.99244 
.99240 


.12278 
.12308 
.12338 
.12367 
.12397 


8.14435 
8.12481 
8.10536 
8.08600 
8.06674 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.10597 
.10626 
.10655 
.10684 
.10713 


.99437 
.99434 
.99431 
.99428 
.99424 


.10657 
.10687 
.10716 
.10746 
.10775 


9.38307 
9.35724 
9.33155 
9.30599 
9.28058 


.12331 
.12360 
.12389 
.12418 
.12447 


.99237 
.99233 
.99230 
.99226 
.99222 


.12426 
.12456 
.12485 
.12515 
.12544 


8.04756 
8.02848 
8.00948 
7.99058 
7.97176 


55 
54 
53 
52 
51 


10 

11 

12 
13 

14 


.10742 
.10771 
.10800 
.10829 
.10858 


.99421 
.99418 
.99415 
.99412 
.99409 


.10805 
.10834 
.10863 
.10893 
.10922 


9.25530 
9.23016 
9.20516 
9.18028 
9.15554 


.12476 
.12504 
.12533 
.12562 
.12591 


.99219 
.99215 
.99211 
.99208 
.99204 


.12574 
.12603 
.12633 
.12662 
.12692 


7.95302 
7.93438 
7.91582 
7.89734 
7.87895 


50 

49 
48 
47 
46 


15 
16 
17 
18 

19 


.10887 
.10916 
.10945 
.10973 
.11002 


.99406 
.99402 
.99399 
.99396 
.99393 


.10952 
.10981 
.11011 
.11040 
.11070 


9.13093 
9.10646 
9.08211 
9.05789 
9.03379 
9.00983 
8.98598 
8.96227 
8.93867 
8.91520 


.12620 
.12649 
.12678 
.12706 
.12735 
.12764 
.12793 
.12822 
.12851 
.12880 


.99200 
.99197 
.99193 
.99189 
.99186 

.99182 
.99178 
.99175 
.99171 
.99167 


.12722 
.12751 
.12781 
.12810 
.12840 
.12869 
.12899 
.12929 
.12958 
.12988 


7.86064 
7.84242 
7.82428 
7.80622 
7.78825 
7.77035 
7.75254 
7.73480 
7.71715 
7.69957 


45 
44 
43 
42 


30 

21 
22 
23 
24 


.11031 
.11060 
.11089 
.11118 
.11147 


.99390 
.99386 
.99383 
.99380 
.99377 


.11099 
.11128 
.11158 
.11187 
.11217 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.11176 
.11205 
» 11234 
.11263 
.11291 


.99374 
.99370 
.99367 
.99364 
.99360 


.11246 
.11276 
.11305 
.11335 
.11364 


8.89185 
8.86862 
8.84551 

8.82252 
8.79964 


.12908 
.12937 
.12966 
.12995 
.13024, 


.99163 
.99160 
.99156 
.99152 
.99148 


.13017 
.13047 
.13076 
.13106 
.13136 
.13165 
.13195 
.13224 
.13254 
.13284 


7.68208 
7.66466 
7.64732 
7.63005 
7.61287 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.11320 
.11349 
.11378 
.11407 
.11436 


.99357 
.99354 
.99351 
.99347 
.99344 


.11394 
.11423 
.11452 
.11482 
.11511 


8.77689 
8.75425 
8.73172 
8.70931 
8.68701 


.13053 
.13081 
.13110 
.13139 
.13168 


.99144 
.99141 
.99137 
.99133 
.99129 


7.59575 
7.57872 
7.56176 
7.54487 
7.52806 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.11465 
.11494 
.11523 
.11552 
.11580 


.99341 
.99337 
.99334 
.99331 
.99327 


.11541 
.11570 
.11600 
.11629 
.11659 


8.66482 
8.64275 
8.62078 
8.59893 
8.57718 


.13197 
.13226 
.13254 
.13283 
.13312 


.99125 
.99122 
.99118 
.99114 
.99110 


.13313 
.13343 
.13372 
.13402 
.13432 


7.51132 
7.49465 
7.47806 
7.46154 
7.44509 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.11609 
.11638 
•11667 
•11696 
.11725 


.99324 
.99320 
.99317 
.99314 
.99310 


.11688 
.11718 
.11747 
.11777 
.11806 


8.55555 
8.53402 
8.51259 
8.49128 
8.47007 


.13341 
.13370 
.13399 
.13427 
.13456 


.99106 
.99102 
.99098 
.99094 
.99091 


.13461 
.13491 
.13521 
.13550 
.13580 


7-42871 
7.41240 
7.39616 
7.37999 
7.36389 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.11754 
•11783 
•11812 
•11840 
.11869 


.99307 
.99303 
.99300 
.99297 
.99293 


.11836 
.11865 
.11895 
.11924 
.11954 


8-44896 
8.42795 
8.40705 
8.38625 
8.36555 


.13485 
.13514 
.13543 
.13572 
.13600 


.99087 
.99083 
.99079 
.99075 
.99071 


.13609 
.13639 
.13669 
.13698 
.13728 


7.34786 
7.33190 
7.31600 
7.30018 
7.28442 


15 
14 
13 
12 
11 


60 

51 
52 
53 
54 


.11898 
.11927 
.11956 
.11985 
.12014 


.99290 
.99286 
.99283 
.99279 
.99276 


.11983 
.12013 
.12042 
.12072 
.12101 


8.34496 
8.32446 
8.30406 
8.28376 
8.26355 


.13629 
.13658 
.13687 
.13716 
.13744 


.99067 
.99063 
.99059 
.99055 
.99051 


.13758 
.13787 
.13817 
.13846 
.13876 


7.26873 
7.25310 
7.23754 
7.22204 
7.20661 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.12043 
.12071 
.12100 
.12129 
.12158 


.99272 
.99269 
.99265 
•99262 
.99258 


.12131 
.12160 
.12190 
.12219 
.12249 


8.24345 
8.22344 
8.20352 
8.18370 
8.16398 


.13773 
.13802 
.13831 
.13860 
.13889 


.99047 
.99043 
99039 
.99035 
.99031 
•99027 


.13906 
.13935 
.13965 
.13995 
.14024 
.14054 


7.19125 
7.17594 
7.16071 
7.14553 
7.13042 


5 
4 
3 
2 
1 


go. 


.12187 


.99255 


.12278 


8.14435 


.13917 


7-11537 


o 


/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


t- 



6dS 



88" 



-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
8^ 9° 



9 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


# 





.13917 


.99027 


.14054 


7.11537 


-15643 


.98769 


-15838 


6.31375 


60 


1 


.13946 


.99023 


.14084 


7.10038 


.15672 


.98764 


.15868 


6-30189 


59 


2 


.13975 


.99019 


.14113 


7.08546 


.15701 


.98760 


.15898 


6.29007 


58 


8 


.14004 


.99015 


.14143 


7.07059 


.15730 


.98755 


.15928 


6.27829 


57 


4 


.14033 


.99011 


.14173 


7.05579 


.15758 


•98751 


.15958 


6.26655 


56 


5 


.14061 


.99006 


.14202 


7.04105 


.15787 


.98746 


.15988 


6.25486 


55 


6 


.14090 


.99002 


.14232 


7.02637 


.15816 


.98741 


.16017 


6.24321 


54 


7 


.14119 


.98998 


.14262 


7.01174 


.15845 


.98737 


.16047 


6.23160 


53 


8 


.14148 


.98994 


.14291 


6.99718 


.15873 


.98732 


.16077 


6.22003 


52 


9 


.14177 


.98990 


.14321 


6.98268 


15902 


.98728 


.16107 


6.20851 


51 


10 


.14205 


.98986 


.14351 


6.96823 


.15931 


.98723 


.16137 


6.19703 


50 


11 


.14234 


.98982 


.14381 


6-95385 


.15959 


.98718 


.16167 


6.18559 


49 


1^ 


.14263 


.98978 


.14410 


6-93952 


.15988 


.98714 


.16196 


6.17419 


48 


13 


.14292 


.98973 


.14440 


6.92525 


.16017 


.98709 


.16226 


6.16283 


47 


14 


.14320 


.98969 


.14470 


6^91104 


•16046 


.98704 


.16256 


6.15151 


46 


15 


.14349 


.98965 


•14499 


6-89688 


.16074 


-98700 


.16286 


6.14023 


45 


16 


.14378 


.98961 


.14529 


6-88278 


•16103 


-98695 


.16316 


6.12899 


44 


17 


.14407 


.98957 


.14559 


6.86874 


•16132 


.98690 


.16346 


6.11779 


43 


18 


.14436 


.98953 


.14588 


6-85475 


•16160 


.98686 


.16376 


6.10664 


42 


19 


.14464 


.98948 


•14618 


6-84082 


•16189 


.98681 


•16405 


6.09552 


41 


30 


.14493 


.98944 


•14648 


6-82694 


•16218 


.98676 


.16435 


6.08444 


40 


21 


.14522 


.98940 


.14678 


6^81312 


•16246 


-98671 


.16465 


6.07340 


39 




.14551 


.98936 


.14707 


6.79936 


•16275 


.98667 


.16495 


6.06240 


38 


tJ8 


.14580 


.98931 


.14737 


6.78564 


.16304 


.98662 


.16525 


6.05143 


37 




.14608 


.98927 


.14767 


6.77199 


•16333 


.98657 


.16555 


6-04051 


36 


«5 


.14637 


•98923 


.14796 


6.75838 


.16361 


.98652 


.16585 


6.02962 


35 


^6 


.14666 


.98919 


.14826 


6.74483 


.16390 


.98648 


.16615 


6.01878 


34 


87 


.14695 


.98914 


.14856 


6.73133 


.16419 


.98643 


.16645 


6.00797 


33 


|28 


.14723 


.98910 


.14886 


6.71789 


.16447 


.98638 


.16674 


5.99720 


32 


29 


.14752 


.98906 


.14915 


6.70450 


.16476 


.98633 


.16704 


5-98646 


31 


«0 


.14781 


•98902 


.14945 


6-69116 


.16505 


.98629 


.16734 


5-97576 


30 


(81 


.14810 


.98897 


.14975 


6.67787 


.16533 


.98624 


.16764 


5-96510 


29 


'8J^ 


.14838 


.98893 


.15005 


6-66463 


•16562 


.98619 


•16794 


5.95448 


28 


^3 


.14867 


.98889 


.15034 


6-65144 


•16591 


.98614 


.16824 


5.94390 


27 


^34 


.14896 


.98884 


.15064 


6.63831 


.16620 


.98609 


•16854 


5.93335 


28 


i85 


.14925 


.98880 


.15094 


6.62523 


•16648 


.98604 


.16884 


5.92283 


25 


8fi 


.14954 


.98876 


.15124 


6.61219 


•16677 


.98600 


.16914 


5.91236 


24 


87 


.14982 


.98871 


.15153 


6.59921 


-16706 


.98595 


.16944 


5.90191 


23 


|88 


.15011 


.98867 


.15183 


6.58627 


•16734 


-98590 


.16974 


5.89151 


22 


I39 


.15040 


.98863 


.15213 


6.57339 


•16763 


.98585 


•17004 


5-88114 


21 


I40 


.15069 


.98858 


.15243 


6.56055 


.16792 


.98580 


.17033 


5-87080 


30 


41 


.15097 


.98854 


.15272 


6.54777 


.16820 


.98575 


.17063 


5.86051 


19 


42 


.15126 


.98849 


.15302 


6.53503 


.16849 


.98570 


.17093 


5.85024 


18 


43 


.15155 


.98845 


.15332 


6.52234 


.16878 


.98565 


.17123 


5.84001 


17 


.44 


.15184 


.98841 


.15362 


6.50970 


.16906 


.98561 


.17153 


5.82982 


16 


145 


.15212 


.98836 


.15391 


6.49710 


.16935 


.98556 


.17183 


5.81966 


15 


146 


.15241 


•98832 


.15421 


6.48456 


.16964 


.98551 


.17213 


5.80953 


14 


47 


.15270 


•98827 


.15451 


6.47206 


•16992 


.98543 


.17243 


5.79944 


13 


Us 


.15299 


•98823 


.15481 


6.45961 


•17021 


.98541 


.17273 


5.78938 


12 


49 


.15327 


•98818 


•15511 


6.44720 


.17050 


-98536 


.17303 


5-77936 


11 


j50 


.15356 


.98814 


•15540 


6.43484 


-17078 


-98531 


.17333 


5.76937 


10 


51 


.15385 


•98809 


.15570 


6.42253 


-17107 


.98526 


.17363 


5.75941 


9 


52 


.15414 


•98805 


.15600 


6.41026 


-17136 


•98521 


.17393 


5-74949 


8 


53 


.15442 


.98800 


.15630 


6.39804 


•17164 


•98516 


.17423 


5.73960 


7 


54 


.15471 


.98796 


•15660 


6-38587 


.17193 


•98511 


.17453 


5.72974 


6 


55 


.15500 


.98791 


.15689 


6.37374 


•17222 


.98506 


.17483 


5.71992 


5 


,56 


.15529 


•98787 


.15719 


6.36165 


•17250 


.98501 


.17513 


5.71013 


4 


'57 


.15557 


•98782 


.15749 


6.34961 


•17279 


.98496 


.17543 


5.70037 


3 


58 


.15586 


•98778 


.15779 


6.33761 


•17308 


.98491 


.17573 


5-69064 


2 


59 


.15615 


.98773 


•15809 


6.32566 


•17336 


•98486 


.17603 


5.68094 


1 


60 


.15643 


.98769 


•15838 


6-31375 


-17365 


.98481 


-17633 


5.67128 


q 


/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


v^ 



81» 



80" 



TABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
10° 11° 


# 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


t 

"60 

59 
58 
57 
56 
55 
54 
53 
52 
51 

50 

49 
48 

47 
46 
45 
44 
43 
42 

40 

39 
38 
37 
36 

35 

34 
33 
32 
31 




1 
2 
3 

4 


.17365 
.17393 
.17422 
.17451 
•17479 


.98481 
.98476 
.98471 
.98466 
.98461 


.17633 
.17663 
.17693 
•17723 
•17753 


5.67128 
5.66165 
5-65205 
5.64248 
5.63295 


.19081 
.19109 
.19138 
•19167 
.19195 


.98163 
.98157 
.98152 
•98146 
-98140 


•19438 
•19468 
•19498 
•19529 
•19559 


5-14455 
5^13658 
5^12862 
5-12069 
5-11279 


5 
6 
7 
8 
9 


.17508 
.17537 
.17565 
.17594 
.17623 


.98455 
.98450 
.98445 
.98440 
.98435 


.17783 
.17813 
.17843 
.17873 
.17903 


5.62344 
5.61397 
5.60452 
5.59511 
5.58573 


.19224 
•19252 
•19281 
•19309 
.19338 


.98135 
-98129 
-98124 
-98118 
98112 


.19589 
•19619 
•19649 
.19680 
•19710 


5-10490 
5-09704 
5-08921 
5-08139 
5-07360 


10 

11 
12 
13 
14 


.17651 
.17680 
.17708 
.17737 
.17766 


.98430 
.98425 
.98420 
.98414 
.98409 


17933 
.17963 
.17993 
.18023 
.18053 


5.57638 
5.56706 
5.55777 
5.54851 
5.53927 


•19366 
•19395 
.19423 
•19452 
•19481 


-98107 
•98101 
-98096 
-98090 
-98084 


•19740 
.19770 
•19801 
•19831 
•19861 


5-06584 
5-05809 
5-05037 
5-04267 
5-03499 


15 
16 
17 
18 
19 


.17794 
.17823 
.17852 
.17880 
.17909 

17937 
.17966 
.17995 
.18023 
.18052 


•98404 
.98399 
.98394 
.98389 
•98383 


.18083 
.18113 
.18143 
.18173 
.18203 


5.53007 
5.52090 
5.51176 
5.50264 
5.49356 


•19509 
.19538 
•19566 
•19595 
-19623 


-98079 
-98073 
-98067 
.98061 
.98056 


.19891 
.19921 
.19952 
•19982 
-20012 


5-02734 
5-01971 
5-01210 
5-00451 
4-99695 


20 

21 
22 
23 

24 


•98378 
.98373 
.98368 
.98362 
•98357 


.18233 
.18263 
•18293 
•18323 
•18353 


5.48451 
5.47548 
5.46648 
5.45751 
5.44857 


-19652 
-19680 
-19709 
-19737 
-19766 


.98050 
•98044 
•98039 
.98033 
.98027 


•20042 
-20073 
•20103 
•20133 
•20164 


4-98940 
4-98188 
4-97438 
4-96690 
4.95945 


)25 

27 
28 
29 


.18081 
.18109 
.18138 
•18166 
.18195 


•98352 
.98347 
.98341 
.98336 
.98331 


•18384 
•18414 
• 18444 
.18474 
.18504 


5.43966 
5.43077 
5.42192 
5.41309 
5-40429 


•19794 
•19823 
-19851 
-19880 
-19908 


.98021 
•98016 
.98010 
.98004 
-97998 


.20194 
•20224 
•20254 
•20285 
-20315 


4.95201 
4.94460 
4.93721 
4.92984 
4-92249 


30 

31 
32 
33 

34 


•18224 
.18252 
.18281 
.18309 
.18338 


•98325 
•98320 
.98315 
.98310 
.98304 


.18534 
•18564 
•18594 
•18624 
.18654 


5.39552 
5.38677 
5.37805 
5.36936 
5. 36070 


-19937 
-19965 
-19994 
-20022 
-20051 


-97992 
.97987 
.97981 
.97975 
-97969 


-20345 
•20376 
•20406 
-20436 
-2046j6 


4-91516 
4-90785 
4-90056 
4-89330 
4-88605 


30 

29 
28 

27 
26 
25 
24 
23 
22 
21 


35 
36 
37 
38 
39 


.18367 
.18395 
.18424 
.18452 
.18481 


.98299 
.98294 
.98288 
.982»3 
•98277 


.18684 
.18714 
.18745 
•18775 
.18805 


5.35206 
5.34345 
5-33487 
5.32631 
5.31778 


•20079 
.20108 
•20136 
•20165 
-20193 


-97963 
.97958 
.97952 
•97946 
-97940 


•20497 
•20527 
•20557 
•20588 
-20618 


4-87882 
4-87162 
4-86444 
4-85727 
4. 85013 


40 

41 
42 
43 

44 


.18509 
•18538 
.18567 
.18595 
.18624 


.98272 
.98267 
.98261 
.98256 
.98250 


.18835 
.18865 
.18895 
.18925 
•18955 


5.30928 
5.30080 
5.29235 
5.28393 
5.27553 


-20222 
•20250 
•20279 
-20307 
-20336 


-97934 
•97928 
-97922 
•97916 
-97910 


-20648 
-20679 
•20709 
•20739 
-20770 


4-84300 
4-83590 
4.82882 
4-82175 
4-81471 


30 

19 , 
18 i 
17 

16 


45 
46 
47 
48 
49 


.18652 
.18681 
.18710 
.18738 
.18767 


.98245 
.98240 
.98234 
•98229 
.98223 


•18986 
•19016 
•19046 
•19076 
•19106 


5.26715 
5.25880 
5.25048 
5.24218 
5.23391 


-20364 
-20393 
-20421 
•20450 
-20478 


-97905 
-97899 
-97893 
-97887 
-97881 


-20800 
-20830 
-20861 
•20891 
•20921 


4-80769 
4-80068 
4-79370. 
4-78673 
4-77978 


15 
14 
13 ; 
12 
11 , 


50 

51 
52 
53 
54 


.18795 
.18824 
.18852 
• 18881 
•18910 


.98218 
.98212 
•98207 
•98201 
.98196 


•19136 
•19166 
•19197 
•19227 
•19257 


5.22566 
5.21744 
5-20925 
5.20107 
5.19293 


-20507 
-20535 
-20563 
-20592 
-20620 


-97875 
-97869 
-97863 
-97857 
-97851 


•20952 
-20982 
-21013 
-21043 
-21073 


4^77286 
4^76595 
4-75906 
4-75219 
4-74534 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.18938 
.18967 
.18995 
.19024 
.19052 


.98190 
.98185 
•98179 
•98174 
•98168 


.19287 
•19317 
•19347 
.19378 
.19408 


5-18480 
5-17671 
5-16863 
5-16058 
5-15256 


• 20649 
-20677 
•20706 
-20734 
•20763 


-97845 
-97839 
-97833 
-97827 
-97821 


-21104 
-21134 
-21164 
-21195 
-21225 


4-73851 
4-73170 
4-72490 
4-71813 
4-71137 


5 1 

l! 

1 


60 


.19081 


•98163 


•19438 


5-14455 


•20791 


-97815 


-21256 


4-70463 


_^l 


^■T- 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 




) 




71 


9* 


70 





7 


8° 







TABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS, AND COTANGENIS. 
12° 13° 


I ' 


Sin. 


Cos. 


Tan. 1 Cot. | 


Sin. Cos. 


Tan. 


Cot. 


' 


( 

\l 

I 3 

i 5 

I 6 
[ 7 

: 8 

( 9 
flO 

811 
U2 
S13 
a 14 

as 

£16 

a? 

118 
J 19 


.20791 
.20820 
.20848 
.20877 
•20905 


.97815 
.97809 
.97803 
.97797 
.97791 


.21256 
.21286 
.21316 
.21347 
.21377 


4 
4 
4 
4 
4 


70463 
69791 
69121 
68452 
67786 


.22495 1 
.22523 1 
.22552 1 
•22580 
•22608 ! 


.97437 
.97430 
.97424 
.97417 
.97411 


.23087 
.23117 
.23148 
.23179 
•23209 


4 
4 
4 
4 
4 


33148 
32573 
32001 
31430 
30860 


60 

59 
58 
57 
56 


.20933 .97784 
.20962 .97778 
.20990 .97772 
.21019 |. 97766 
.21047 !. 97760 


•21408 
•21438 
•21469 
.21499 
.21529 


4 
4 
4 
4 
4 


67121 
66458 
65797 
65138 
64480 


•22637 
•22665 
•22693 
•22722 
•22750 
•22778 
•22807 
•22835 
•22863 
.22892 


.97404 
•97398 
•97391 
•97384 
•97378 


•23240 
•23271 
.23301 
•23332 
•23363 


4 
4 
4 
4 
4 


30291 
29724 
29159 
28595 
28032 


55 
54 
53 
52 
51 


.21076 
.21104 
.21132 
.21161 
.21189 


.97754 
•97748 
.97742 
.97735 
.97729 


.21560 
•21590 
.21621 
.21651 
.21682 


4 
4 
4 
4 
4 


63825 
63171 
62518 
61868 
61219 


.97371 
•97365 
•97358 
•97351 
•97345 


.23393 
.23424 
.23455 
•23485 
•23516 


4 
4 
4 
4 
4 


27471 
26911 
26352 
25795 
25239 


50 

49 
48 

47 
46 


.21218 
.21246 
.21275 
.21303 
.21331 


.97723 
.97717 
.97711 
•97705 
•97698 


.21712 
.21743 
.21773 
•21804 
•21834 


4 
4 
4 
4 
4 


60572 
59927 
59283 
58641 
58001 


.22920 
.22948 
.22977 
.23005 
•23033 


•97338 
•97331 
•97325 
•97318 
.97311 


•23547 
•23578 
.23608 
•23639 
.23670 


4 
4 
4 
4 
4 


24685 
24132 
23580 
23030 
22481 


45 
44 
43 
42 
41 


£20 

8 21 
122 
3 23 
^24 


•21360 
.21388 
.21417 
.21445 
.21474 


•97692 
.97686 
.97680 
.97673 
.97667 


•21864 
.21895 
•21925 
.21956 
.21986 


4 
4 

t 

4 


57363 
56726 
56091 
55458 
54826 


.23062 
.23090 
.23118 
.23146 ! 
•23175 


.97304 
•97298 
•97291 
•97284 
•97278 


•23700 
.23731 
•23762 
.23793 
.23823 


4 
4 
4 
4 
4 


21933 
21387 
20842 
20298 
19756 


40 

39 
38 
37 
36 


1(25 
?26 
S27 
})28 
fj29 


.21502 
.21530 
.21559 
.21587 
.21616 


.97661 
.97655 
.97648 
.97642 
•97636 


.22017 
.22047 
.22078 
.22108 
.22139 


4 
4 
4 
4 
4 


54196 
53568 
52941 
52316 
51693 


•23203 
•23231 
•23260 
•23288 
.23316 


.97271 
•97264 
.97257 
.97251 
.97244 
.97237 
.97230 
.97223 
.97217 
•97210 
•97203 
•97196 
•97189 
•97182 
•97176 


.23854 
.23885 
.23916 
.23946 
•23977 


4 
4 
4 
4 
4 


19215 
18675 
18137 
17600 
17064 


35 
34 
33 
32 
31 


iso 

J 31 
32 
33 

3 34 


.21644 
.21672 
•21701 
.21729 
.21758 


•97630 
.97623 
.97617 
.97611 
•97604 


•22169 
•22200 
•22231 
.22261 
•22292 


4 
4 
4 
4 
4 


51071 
50451 
49832 
49215 
48600 


.23345 
•23373 
.23401 
.23429 
•23458 


•24008 
•24039 
•24069 
•24100 
•24131 


4 
4 
4 
4 
4 


16530 
15997 
15465 
14934 
14405 


30 

29 
28 
27 
26 


^35 
: 36 

:37 

38 

1 39 


.21786 
.21814 
.21843 
.21871 
•21899 


.97598 
.97592 
.97585 
.97579 
•97573 


.22322 
.22353 
.22383 
.22414 
• 22444 


4 
4 
4 
4 
4 


47986 
47374 
46764 
46155 
45548 


23486 
.23514 
.23542 
.23571 
.23599 


•24162 
.24193 
.24223 
.24254 
.24285 


4 
4 
4 
4 
4 


13877 
13350 
12825 
12301 
11778 


25 
24 
23 
22 
21 


Uo 

41 
42 
43 
44 


.21928 
.21956 
.21985 
.22013 
.22041 


•97566 
•97560 
.97553 
.97547 
.97541 


•22475 
•22505 
•22536 
•22567 
.22597 


4 
4 
4 
4 
4 


44942 
44338 
43735 
43134 
42534 


.23627 
.23656 
.23684 
•23712 
.23740 


.97169 
.97162 
•97155 
•97148 
.97141 


.24316 
.24347 
.24377 
• 24408 
•24439 


4 
4 
4 
4 
4 


11256 
10736 
10216 
09699 
09182 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•22070 
.22098 
•22126 
.22155 
•22183 


.97534 
.97528 
.97521 
.97515 
.97508 


•22628 
•22658 
•22689 
•22719 
•22750 


4 
4 
4 
4 
4 


41936 
41340 
40745 
40152 
39560 


•23769 
.23797 
.23825 
.23853 
.23882 


.97134 
•97127 
•97120 
•97113 
•97106 


•24470 
•24501 
•24532 
•24562 
.24593 


4 
4 
4 
4 
4 


08666 
08152 
07639 
07127 
06616 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•22212 
•22240 
.22268 
.22297 
.22325 


.97502 
.97496 
.97489 
.97483 
•97476 


•22781 
.22811 
•22842 
•22872 
•22903 


4 
4 
4 
4 
4 


38969 
38381 
37793 
37207 
36623 


•23910 
•23938 
•23966 
.23995 
•24023 


•97100 
•97093 
•97086 
.97079 
.97072 


.24624 
.24655 
.24686 
•24717 
•24747 


4 
4 
4 
4 
4 


06107 
05599 
05092 
04586 
04081 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.22353 
.22382 
•22410 
.22438 
•22467 . 


•97470 
•97463 
•97457 
.97450 : 
•97444 > 


•22934 
•22964 
.22995 
.23026 
.23056 


4 
4 
4 
4. 
4 


36040 
35459 
34879 
34300 
33723 


.24051 
.24079 
.24108 
.24136 
.24164 


.97065 
.97058 
.97051 
.97044 
•97037 


•24778 
•24809 
• 24840 
•24871 
.24902 


4 
4 
4 
4 
4 


03578 
03076 
02574 
02074 
01576 


5 
4 
3 

2 

1 


60 


.22495 


•97437 


•23087 


4 


33148 


.24192 


.97030 


.24933 


4 


01078 





' 


Cos. 


Sin. 1 Got. 


Tan. 1 


Cos. 


Sin. 


Cot. 


Tan. 





77° 



701 



76° 



TABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
14° 15° 


T/ 


f 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 






1 
2 
3 
4 


.24192 
.24220 
.24249 
.24277 
.24305 


.97030 
•97023 
.97015 
•97008 
.97001 


•24933 
.24964 
•24995 
•25026 
•25056 


4.01078 
4.00582 
4.00086 
3.99592 
3.99099 


-25882 
•25910 
•25938 
.25966 
.25994 


-96593 
.96585 
-96578 
-96570 
-96562 


•26795 
•26826 
•26857 
•26888 
.26920 


3-73205 
3-72771 
3-72338 
3-71907 
3-71476 


59 

58 
57 
58 




5 
6 
7 
8 
9 


.24333 
.24362 
.24390 
.24418 
. 24446 


.96994 
.96987 
.96980 
.96973 
.96966 


•25087 
•25118 
.25149 
.25180 
.25211 


3-98607 
3.98117 
3-97627 
3-97139 
3.96651 


•26022 
.26050 
•26079 
.26107 
•26135 


.965^5 
.96547 
.96540 
.96532 
•96524 


.26951 
•26982 
•27013 
•27044 
.27076 


3-71046 
3-70616 
3. 70188 
3.69761 
3-69335 


55 

54' 
53 
52 
51 i 
50 
49 
48 
47 
48 
45 
44, 
43 
42 
-ii 
40 
39 
38 
37 
38 




10 

11 
12 
13 
14 


. 24474 
.24503 
•24531 
.24559 
.24587 
.24615 
. 24644 
.24672 
.24700 
.24728 


.96959 
.96952 
.96945 
.96937 
.96930 


•25242 
•25273 
•25304 
•25335 
•25366 


3-96165 
3-95680 
3.95196 
3.94713 
3.94232 


•26163 
.26191 
•26219 
•26247 
•26275 


•96517 
.96509 
•96502 
•96494 
.96486 


-27107 
-27138 
.27169 
-27201 
-27232 


3-68909 
3-68485 
3-68061 
3.67638 
3-67217 




15 
16 
17 
18 
19 


.96923 
.96916 
.96909 
.96902 
•96894 

.96887 
.96880 
.96873 
•96866 
•96858 


•25397 
•25428 
•25459 
•25490 
.25521 


3-93751 
3-93271 
3.92793 
3-92316 
3. 91839 


•26303 
.26331 
.26359 
.26387 
.26415 


.96479 
.96471 
.96463 
•96456 
.96448 


-27263 
.27294 
•27326 
.27357 
.27388 


3-66796 
3.66376 
3-65957 
3-65538 
3-65121 




30 

21 
22 
23 
24 


.24756 
.24784 
.24813 
.24841 
.24869 

.24897 
.24925 
.24954 
•24982 
.25010 


.25552 
•25583 
.25614 
.25645 
•25676 


3-91364 
3-90890 
3-90417 
3-89945 
3-89474 


• 26443 
.26471 
.26500 
.26528 
.26556 


.96440 
.96433 
.96425 
.96417 
•96410 


•27419 
•27451 
•27482 
•27513 
.27545 


3-64705 
3.64289 
3.63874 
3.63461 
3-63048 




25 
26 
27 
28 
29 


•96851 
.96844 
.96837 
•96829 
•96822 


•25707 
•25738 
.25769 
.25800 
•25831 


3.89004 
3^88536 
3.88068 
3.87601 
3.87136 


•26584 
•26612 
•26640 
.26668 
.26698 


•96402 
•96394 
•96386 
.96379 
•96371 


.27576 
•27607 
•27638 
•27670 
•27701 


3-62636 
3^62224 
3^61814 
3-61405 
3 • 60996 


35 

34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 




30 

31 
32 
33 
34 


.25038 
.25066 
25094 
.25122 
.25151 


.96815 
.96807 
.96800 
.96793 
.96786 


•25862 
•25893 
•25924 
•25955 
•25986 
.26017 
.26048 
.26079 
•26110 
•26141 


3^86671 
3 •86208 
3 • 85745 
3. 85284 
3-84824 


.26724 
.26752 
•26780 
•26808 
.26836 


.96363 
-96355 
.96347 
•96340 
•96332 


•27732 
•27764 
•27795 
-27826 
-27858 


3 •60588 
3-60181 
3-59775 
3-59370 
3-58966 




35 
36 
37 
38 
39 


.25179 
.25207 
.25235 
.25263 
.25291 


.96778 
.96771 
.96764 
.96756 
.96749 


3-84364 
3-83908 
3-83449 
3 82992 
3. 82537 


•26864 
•26892 
•26920 
•26948 
•26976 


•96324 
.96316 
.96308 
.96301 
.96293 


-27889 
•27921 
-27952 
•27983 
•28015 


3-58562 
3-58160 
3-57758 
3.57357 
3-56957 




40 

41 
42 
43 
44 


.25320 
•25348 
.25376 
.25404 
.25432 


•96742 
•96734 
•96727 
.96719 
•96712 


•26172 
•26203 
.26235 
.26266 
.26297 


3-82083 
3-81630 
3-81177 
3-80726 
3-80276 


•27004 
•27032 
•27060 
27088 
.27116 


•96285 
.96277 
.96269 
•96261 
•96253 


-28046 
-28077 
-28109 
-28140 
-28172 


3-56557 
3-56159 
3-55761 
3-55364 
3-54968 


20 

19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 




45 
46 
47 
48 
49 


• 25460 
.25488 
.25516 
.25545 
.25573 


•96705 
•96697 
•96690 
96682 
•96675 


.26328 
.26359 
.26390 
.26421 
.26452 


3-79827 
3 •79378 
3.78931 
3-78485 
3-78040 


.27144 
-27172 
-27200 
•27228 
-27256 


•96246 
•96238 
•96230 
•96222 
.96214 


•28203 
•28234 
•28266 
•28297 
•28329 


3-54573 
3-54179 
3-53785 
3.53393 
3-53001 




50 

51 
52 
53 
54 


.25601 
.25629 
.25657 
•25685 
•25713 


•96667 
•96660 
•96653 
•96645 
.96638 


.26483 
.26515 
.26546 
.26577 
.26608 
.26639 
.26670 
.26701 
.26733 
•26764 


3-77595 
3-77152 
3-76709 
3-76268 
3-75828 


-27284 
-27312 
.27340 
.27368 
.27396 


•96206 
•96198 
.96190 
•96182 
.96174 


•28360 
•28391 
•28423 
•28454 
.28486 


3-52609 
3-52219 
3.51829 
3.51441 
3.51053 




55 
56 
57 
58 
59 


•25741 
.25769 
.25798 
.25826 
.25854 


•96630 
•96623 
.96615 
.96608 
.96600 
.96593 


3 •75388 
3-74950 
3^74512 
3^74075 
3-73640 
3-73205 


.27424 
.27452 
.27480 
.27508 
27536 
•27564 


•96166 
•96158 
•96150 
•96142 
.96134 


.28517 
.28549 
.28580 
.28612 
.28643 


3.50666 
3.50279 
3-49894 
3-49509 

3-49125 


5 

4 

2 

1 






60 .25882 


.26795 


.96126 


•28675 
Cot. 


3 • 48741 
Tan. 




' Cos. 


Sin. 1 Cot. 


Tan. 


Cos. 


Sin. 








i 


rs^ 


70 


2 


i 


740 


( 







TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 



L 




16° 






1 


7° 






if 

c 4 


Sin, 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


f 


.27564 
.27592 
.27620 
.27648 
.27676 


.96126 
.96118 
.96110 
.96102 
.96094 


.28675 
.28706 
.28738 
.28769 
.28800 


3.48741 
3-48359 
3.47977 
3.47596 
3.47216 


-29237 
-29265 
-29293 
-29321 
-29348 


-95630 
-95622 
-95613 
-95605 
-95596 


-30573 
-30605 
-30637 
-30669 
.30700 


3-27085 
3-26745 
3-26406 
3-26067 
3-25729 


60 

59 
58 
57 
56 


« 5 

6 

1 7 
8 

^ 9 


.27704 
.27731 
.27759 
.27787 
.27815 


.96086 
.96078 
.96070 
.96062 
.96054 


.28832 
.28864 
.28895 
.28927 
.28958 


3-46837 
3-46458 
3-46080 
3.45703 
3.45327 


-29376 
-29404 
-29432 
-29460 
.29487 


-95588 
-95579 
.95571 
.95562 
-95554 


.30732 
.30764 
.30796 
.30828 
.30860 


3-25392 
3-25055 
3-24719 
3-24383 
3 - 24049 


55 
54 
53 
52 
51 


t 10 

11 
-12 

U4 


.27843 
.27871 
.27899 
.27927 
.27955 


.96046 
.96037 
.96029 
.96021 
.96013 


.28990 
.29021 
.29053 
.29084 
.29116 


3.44951 
3.44576 
3.44202 
3.43829 
3.43456 


.29515 
.29543 
.29571 
-29599 
-29626 


.95545 
.95536 
.95528 
.95519 
-95511 


.30891 
.30923 
.30955 
.30987 
.31019 


3-23714 
3-23381 
3-23048 
3.22715 
3-22384 


50 

49 
48 
47 
46 


^15 

18 
-19. 


.27983 
.28011 
.28039 
.28067 
.28095 


.96005 
.95997 
.95989 
.95981 
.95972 


.29147 
.29179 
.29210 
.29242 
.29274 


3.43084 
3.42713 
3.42343 
3.41973 
3.41604 


-29654 
.29682 
.29710 
.29737 
.29765 


-95502 
-95493 
-95485 
-95476 
.95467 


.31051 
.31083 
.31115 
.31147 
-31178 


3-22053 
3 21722 
3-21392 
3-21063 
3-20734 


45 
44 
43 
42 
41 


^20 

21 

j22 

J 24 


.28123 
.28150 
.28178 
•28206 
.28234 


.95964 
.95956 
.95948 
.95940 
.95931 


.29305 
.29337 
.29368 
.29400 
.29432 


3.41236 
3.40869 
3.40502 
3.40136 
3.39771 


.29793 
.29821 
.29849 
.29876 
-29904 


-95459 
.95450 
.95441 
.95433 
.95424 


.31210 
.31242 
.31274 
.31306 
•31338 


3 . 20406 
3.20079 
3.19752 
3.19426 
3-19100 


40 

39 
38 
37 
36 


(25 

26 

27 

^28 

;29 


.28262 
.28290 
.28318 
.28346 
.28374 


.95923 
.95915 
.95907 
.95898 
.95890 


.29463 
.29495 
.29526 
.29558 
.29590 


3.39406 
3-39042 
3.38679 
3.38317 
3-37955 


-29932 
.29960 
.29987 
.30015 
.30043 


.95415 
.95407 
.95398 
.95389 
-95380 


.31370 
-31402 
-31434 
-31466 
-31498 


3-18775 
3.18451 
3-18127 
3-17804 
3-1748] 


35 
34 
33 
32 
31 


Uo 

,31 

33 
34 


.28402 
.28429 
•28457 
•28485 
.28513 


.95882 
.95874 
.95865 
.95857 
.95849 


.29621 
.29653 
.29685 
.29716 
.29748 


3.37594 
3.37234 
3.36875 
3.36516 
3.36158 


-30071 
.30098 
-30126 
-30154 
-30182 


-95372 
-95363 
-95354 
-95345 
-95337 


.31530 
.31562 
.31594 
.31626 
-31658 


3-17159 
3-16838 
3.16517 
3.16197 
3-15877 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.28541 
•28569 
•28597 
.28625 
.28652 
.28680 
.28708 
.28736 
.28764 
.28792 


.95841 
.95832 
.95824 
.95816 
.95807 
.95799 
.95791 
.95782 
.95774 
.95766 


.29780 
.29811 
.29843 
.29875 
.29906 

.29938 
.29970 
.30001 
.30033 
.30065 


3.35800 
3.35443 
3.35087 
3.34732 
3. 34377 


.30209 
.30237 
.30265 
.30292 
-30320 


-95328 
-95319 
-95310 
•95301 
.95293 


-31690 
-31722 
.31754 
.31786 
-31818 


3-15558 
3-15240 
3-14922 
3-14605 
3-14288 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


3.34023 
3.33670 
3.33317 
3-32965 
3.32614 


-30348 
-30376 
-30403 
-30431 
-30459 


-95284 
-95275 
-95266 
-95257 
-95248 


-31850 
.31882 
-31914 
-31946 
-31978 


3-13972 
3-13656 
3.13341 
3-13027 
3-12713 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.28820 
.28847 
.28875 
.28903 
.28931 


•95757 
.95749 
.95740 
•95732 
.95724 


.30097 
.30128 
.30160 
.30192 
.30224 


3.32264 
3.31914 
3.31565 
3.31216 
3.30868 


-30486 
.30514 
.30542 
.30570 
.30597 


-95240 
-95231 
-95222 
-95213 
-95204 


.32010 
-32042 
.32074 
.32106 
-32139 


3-12400 
3.12087 
3.11775 
3-11464 
3-11153 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.28959 
.28987 
.29015 
.29042 
.29070 


.95715 
.95707 
.95698 
.95690 
.95681 


.30255 
.30287 
.30319 
.30351 
.30382 


3.30521 
3.30174 
3.29829 
3-29483 
3-29139 


.30625 
.30653 
.30680 
.30708 
-30736 


-95195 
.95186 
.95177 
.95168 
-95159 


-32171 
.32203 
-32235 
32267 
32299 
32331 
32363 
32396 
32428 
32460 


3.10842 
3.10532 
3-10223 
3.09914 
3-09606 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.29098 
.29126 
.29154 
.29182 
.29209 


.95673 
.95664 
.95656 
.95647 
•95639 


.30414 
.30446 
.30478 
.30509 
.30541 


3-28795 
3.28452 
3.28109 
3-27767 
3-27426 


.30763 
•30791 
.30819 
.30846 
.30874 


-95150 
.95142 
.95133 
.95124 
95115 


3-09298 
3-08991 
3.08685 
3-08379 
3-08073 


5 

4 
3 
2 
1 


go. 


.29237 


.95630 


.30573 


3-27085 


-30902 


95106 


32492 


3-07^68 





; ' 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. Cot. 1 


Tan. 


/ 



73° 



703^ 



73° 



TABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
18° 19° 


t 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 
4 


•30902 
.30929 
.30957 
.30985 
.31012 

.31040 
.31068 
.31095 
.31123 
.31151 


.95106 
.95097 
.95088 
.95079 
.95070 


•32492 
•32524 
.32556 
•32588 
.32621 


3-07768 
3.07464 
3.07160 
3.06857 
3.06554 


•32557 
-32584 
•32612 
•32639 
•32667 


•94552 
•94542 
•94533 
.94523 
.94514 


•34433 
•34465 
•34498 
•34530 
•34563 


2-90421 
2-90147 
2 •89873 
2-89600 
2-89327 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.95061 
.95052 
.95043 
.95033 
.95024 


.32653 
.32685 
.32717 
.32749 
.32782 
.32814 
•32846 
.32878 
.32911 
.32943 


3.06252 
3.05950 
3.05649 
3-05349 
3-05049 


•32694 
•32722 
•32749 
•32777 
-32804 


.94504 
.94495 
.94485 
.94476 
. 94466 


.34596 
.34628 
.34661 
.34693 
.34726 


2-89055 
2.88783 
2.88511 
2-88240 
2-87970 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


.31178 
.31206 
.31233 
•31261 
.31289 
•31316 
.31344 
•31372 
•31399 
•31427 
•31454 
•31482 
•31510 
.31537 
.31565 


.95015 
.95006 
.94997 
.94988 
.94979 


3-04749 
3-04450 
3-04152 
3-03854 
3-03556 


•32832 
-32859 
-32887 
•32914 
.32942 


.94457 
.94447 
.94438 
.94428 
.94418 


.34758 
.34791 
•34824 
•34856 
•34889 
•34922 
.34954 
.34987 
.35020 
•35052 


2.87700 
2.87430 
2.87161 
2.86892 
2.86624 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.94970 
.94961 
.94952 
.94943 
.94933 

.94924 
.94915 
.94906 
.94897 
.94888 


.32975 
.33007 
.33040 
.33072 
.33104 


3- 03260 
3-02963 
3-02667 
3-02372 
3.02077 


.32969 
.32997 
.33024 
.33051 
•33079 


.94409 
.94399 
.94390 
.94380 
.94370 


2.86356 
2.86089 
2-85822 
2-85555 
2-85289 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.33136 
•33169 
•33201 
•33233 
.33266 


3-01783 
3-01489 
3-01196 
3-00903 
3-00611 
3-00319 
3-00028 
2-99738 
2-99447 
2-99158 


•33106 
.33134 
.33161 
•33189 
.33216 


.94361 
.94351 
.94342 
.94332 
.94322 


•35085 
.35118 
•35150 
•35183 
•35216 


2-85023 
2.84758 
2.84494 
2-84229 
2-83965 
2-83702 
2-83439 
2-83176 
2-82914 
2-82653 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.31593 
.31620 
.31648 
.31675 
.31703 


.94878 
.94869 
.94860 
.94851 
.94842 


.33298 
.33330 
.33363 
.33395 

.33427 


.33244 
•33271 
•33298 
•33326 
•33353 


.94313 
.94303 
.94293 
.94284 
.94274 


•35248 
.35281 
•35314 
•35346 
.35379 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.31730 
.31758 
.31786 
.31813 
•31841 


.94832 
.94823 
.94814 
.94805 
.94795 


.33460 
•33492 
.33524 
.33557 
.33589 


2-98868 
2-98580 
2-98292 
2-98004 
2-97717 


•33381 
•33408 
•33436 
•33463 
•33490 


.94264 
.94254 
•94245 
•94235 
•94225 


•35412 
•35445 
.35477 
.35510 
.35543 
.35576 
.35608 
.35641 
•35674 
•35707 


2-82391 
2.82130 
2^81870 
2^81610 
2^81350 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.31868 
.31896 
.31923 
.31951 
•31979 


.94786 
.94777 
.94768 
.94758 
.94749 


.33621 
.33654 
•33686 
•33718 
•33751 


2-97430 
2-97144 
2-96858 
2-96573 
2-96288 


•33518 
•33545 
•33573 
-33600 
•33627 


•94215 
•94206 
.94196 
.94186 
•94176 


2.81091 
2.80833 
2^80574 
2^80316 
2.80059 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•32006 
•32034 
•32061 
.32089 
.32116 


.94740 
.94730 
.94721 
.94712 
.94702 


•33783 
•33816 
•33848 
.33881 
•33913 


2^96004 
2-95721 
2-95437 
2-95155 
2.94872 


-33655 
•33682 
•33710 
•33737 
•33764 


.94167 
•94157 
.94147 
.94137 
.94127 


.35740 
.35772 
.35805 
.35838 
•35871 


2-79802 
2.79545 
2-79289 
2-79033 
2-78778 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.32144 
.32171 
.32199 
.32227 
•32254 


.94693 
.94684 
.94674 
.94665 
•94656 


•33945 
•33978 
•34010 
•34043 
•34075 


2^94591 
2.94309 
2.94028 
2.93748 
2-93468 


•33792 
-33819 
-33846 
-33874 
-33901 


•94118 
•94108 
•94098 
-94088 
-94078 


.35904 
.35937 
.35969 
•36002 
•36035 


2-78523 
2-78269 
2-78014 
2^77761 
2-77507 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


•32282 
•32309 
•32337 
•32364 
•32392 


.94646 
.94637 
.94627 
.94618 
.94609 


•34108 
•34140 
•34173 
•34205 
•34238 


2-93189 
2.92910 
2.92632 
2.92354 
2.92076 


-33929 
-33956 
•33983 
•34011 
•34038 


-94068 
-94058 
•94049 
•94039 
•94029 
•94019 
•94009 
•93999 
•93989 
•93979 


•36068 
•36101 
•36134 
•36167 
•36199 


2-77254 
2-77002 
2-76750 
2^76498 
2-76247 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•32419 
•32447 
•32474 
32502 
•32529 


.94599 
.94590 
•94580 
•94571 
•94561 


•34270 
•34303 
34335 
•34368 
.34400 


2.91799 
2.91523 
2.91246 
2-90971 
2-90696 


•34065 
•34093 
-34120 
.34147 
•34175 


•36232 
•36265 
•36298 
36331 
36364 


2-75996 
2-75746 
2.75496 
2.75246 
2.74997 


5 
4 
3 
2 
1 


60 


•32557 


■94552 


•34433 


2-90421 


.34202 


93969 


36397 


2.74748 







(V)s. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot^ 


Tan. 


' 



71° 



704 



70° 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
30° 31° 



t 


Sin. 


Cos. 


Tan. 1 Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




1 

2 
3 

4 

5 
6 
7 
8 
9 


.34202 
.34229 
.34257 
.34284 
.34311 


.93969 
.93959 
.93949 
.93939 
.93929 


.36397 
.36430 
.36463 
.36496 
.36529 


2.74748 
2.74499 
2.74251 
2-74004 
2.73756 


.35837 
.35864 
.35891 
.35918 
.35945 


.93358 
•93348 
.93337 
.93327 
.93316 


.38386 
.38420 
.38453 
.38487 
.38520 


2-60509 
2-60283 
2-60057 
2-59831 
2-59606 


60 

59 
58 
57 
56 


.34339 
•34366 
.34393 
.34421 
.34448 


.93919 
.93909 
.93899 
.93889 
•93879 


.36562 
.36595 
.36628 
.36661 
.36694 


2.73509 
2.73263 
2.73017 
2.72771 
2.72526 


•35973 
•36000 
.36027 
•36054 
•36081 


.93306 
•93295 
.93285 
•93274 
•93264 


•38553 
•38587 
•38620 
•38654 
-38687 


2-59381 
2-59156 
2-58932 
2-58708 
2-58484 


55 
54 
53 
52 
51 


10 

}.l 
12 
13 
14 


.34475 
.34503 
.34530 
.34557 
.34584 


.93869 
.93859 
•93849 
.93839 
.93829 


.36727 
.36760 
.36793 
•36826 
.36859 


2.72281 
2.72036 
2.71792 
2.71548 
2.71305 


•36108 
.36135 
.36162 
•36190 
•36217 


•93253 
•93243 
•93232 
•93222 
.93211 


-38721 
.38754 
.38787 
•38821 
•38854 


2-58261 
2.58038 
2.57815 
2.57593 
2.57371 


50 

49 
48 

47 
46 


15 
16 
17 
18 

19 


.34612 
34639 
.34666 
.34694 
.34721 


.93819 
.93809 
.93799 
.93789 
.93779 


•36892 
•36925 
.36958 
.36991 
.37024 


2-71062 
2.70819 
2-70577 
2-70335 
2-70094 


.36244 
.36271 
•36298 
.36325 
•36352 


.93201 
.93190 
.93180 
.93169 
•93159 


-38888 
-38921 
•38955 
•38988 
•39022 


2.57150 
2.56928 
2.56707 
2.56487 
2.56266 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.34748 
.34775 
.34803 
.34830 
.34857 


.93769 
.93759 
.93748 
.93738 
.93728 


.37057 
.37090 
.37123 
.37157 
.37190 


2-69853 
2.69612 
2.69371 
2.69131 
2^68892 


•36379 
.36406 
.36434 
.36461 
•36488 


.93148 
.93137 
.93127 
.93116 
.93106 


•39055 
-39089 
-39122 
-39156 
•39190 


2.56046 
2.55827 
2.55608 
2.55389 
2.55170 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•34884 
•34912 
•34939 
.34966 
.34993 


.93718 
.93708 
.93698 
.93688 
.93677 


.37223 
.37256 
.37289 
.37322 
.37355 


2-68653 
2.68414 
2.68175 
2.67937 
2.67700 


•36515 
•36542 
.36569 
.36596 
.36623 


.93095 
.93084 
.93074 
.93063 
.93052 


-39223 
39257 
•39290 
.39324 
•39357 


2.54952 
2.54734 
2.54516 
2.54299 
2.54082 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.35021 
.35048 
•35075 
.35102 
.35130 


.93667 
.93657 
•93647 
•93637 
•93626 


.37388 
.37422 
•37455 
•37488 
•37521 


2.67462 
2.67225 
2.66989 
2.66752 
2^66516 


.36650 
.36677 
•36704 
.36731 
•36758 


.93042 
.93031 
•93020 
•93010 
•92999 


-39391 
•39425 
•39458 
•39492 
•39526 
•39559 
-39593 
•39626 
•39660 
•39694 


2^53865 
2^53648 
2.53432 
2.53217 
2.53001 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.35157 
.35184 
.35211 
.35239 
.35266 


.93616 
.93606 
.93596 
•93585 
•93575 


•37554 
•37588 
•37621 
.37654 
.37687 


2 •66281 
2.66046 
2.65811 
2.65576 
2.65342 


•36785 
•36812 
•36839 
•36867 
•36894 


.92988 
•92978 
•92967 
•92956 
•92945 


2.52786 
2.52571 
2.52357 
2.52142 
2.51929 


25 
24 
23 
22 
21 


40 

41 
42 
43 

44 


.35293 
.35320 
•35347 
•35375 
•35402 


.93565 
.93555 
.93544 
.93534 
.93524 


.37720 
.37754 
•37787 
•37820 
.37853 


2.65109 
2.64875 
2.64642 
2.64410 
2.64177 


•36921 
•36948 
•36975 
•37002 
•37029 


.92935 
•92924 
•92913 
•92902 
•92892 


•39727 
•39761 
•39795 
•39829 
•39862 


2.51715 
2.51502 
2.51289 
2.51076 
2.50864 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


•35429 
•35456 
•35484 
•35511 
•35538 


.93514 
.93503 
.93493 
.93483 
.93472 


.37887 
.37920 
•37953 
•37986 
•38020 


2.63945 
2.63714 
2 •63483 
2^63252 
2 •63021 


•37056 
•37083 
•37110 
•37137 
•37164 


•92881 
•92870 
•92859 
.92849 
.92838 


-39896 
•39930 
•39963 
•39997 
•40031 


2.50652 
2.50440 
2.50229 
2^50018 
2^49807 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.35565 
.35592 
•35619 
•35647 
•35674 


.93462 
.93452 
•93441 
.93431 
•93420 


.38053 
.38086 
•38120 
•38153 
•38186 


2-62791 
2-62561 
2-62332 
2 •62103 
2^61874 


.37191 
.37218 
.37245 
•37272 
•37299 


.92827 
.92816 
.92805 
.92794 
•92784 


•40065 
•40098 
•40132 
•40166 
•40200 


2.49597 
2.49386 
2.49177 
2.48967 
2^48758 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•35701 
•35728 
•35755 
•35782 
•35810 


.93410 
.93400 
.93389 
•93379 
.93368 


.38220 
.38253 
•38286 
•38320 
•38353 


2.61646 
2.61418 
2.61190 
2.60963 
2.60736 


•37326 
•37353 
37380 
•37407 
.37434 


•92773 
•92762 
•92751 
•92740 
•92729 


•40234 
•40267 
•40301 
-40335 
-40369 


2.48549 
2-48340 
2-48132 
2^47924 
2.47716 


5 

4 
3 

2 

1 


60 


•35837 


.93358 


•38386 


2.60509 


.37461 


•92718 


-40403 


2.47509 







Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. Tan. | 


/ 



69° 



705 



68° 



TABLE IX.- 


-NATURAL SINES, COSINES. TANGENTS, AND COTANGENTS. 
33° 33° 1 


r 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


m 




1 

2 
3 

4 


•37461 
.37488 
.37515 
.37542 
.37569 


.92718 
.92707 
.92697 
.92686 
.92675 


.40403 
.40436 
.40470 
.40504 
.40538 


2.47509 
2.47302 
2.47095 
2.46888 
2.46682 


-39073 
.39100 
.39127 
.39153 
•39180 


.92050 
.92039 
.92028 
.92016 
.92005 


.42447 
.42482 
.42516 
.42551 
.42585 


2.35585 
2.35395 
2.35205 
2.35015 
2-34825 


^80 

59 
58 
57 
56 

55 
54 
53 
52 
51 

50 

49 
48 
47 
46 
45 
44 
43 
42 
41 


5 
6 
7 
8 
9 


.37595 
.37622 
.37649 
.37676 
.37703 


.92664 
.92653 
.92642 
.92631 
92620 


.40572 
.40606 
.40640 
.40674 
.40707 


2.46476 
2.46270 
2.46065 
2.45860 
2.45655 


-39207 
-39234 
.39260 
.39287 
-39314 


.91994 
.91982 
.91971 
.91959 
.91948 


.42619 
.42654 
.42688 
.42722 
.42757 


2-34636 
2.34447 
2.34258 
2-34069 
2-33881 


10 

11 
12 
13 
14 


.37730 
.37757 
.37784 
•37811 
.37838 


.92609 
.92598 
.92587 
.92576 
.92565 


.40741 
.40775 
.40809 
.40843 
.40877 


2.45451 
2.45246 
2.45043 
2.44839 
2.44636 


-39341 
.39367 
-39394 
.39421 
-39448 


.91936 
.91925 
.91914 
.91902 
.91891 


.42791 
.42826 
.42860 
.42894 
.42929 


2-33693 
2-33505 
2.33317 
2-33130 
2.32943 


15 
16 
17 
18 
19 


.37865 
.37892 
.37919 
.37946 
.37973 


.92554 
.92543 
.92532 
.92521 
.92510 


.40911 
.40945 
.40979 
.41013 
.41047 


2.44433 
2-44230 
2-44027 
2-43825 
2.43623 


-39474 
.39501 
.39528 
.39555 
.39581 


.91879 
-91868 
-91856 
-91845 
•91833 


.42963 
.42998 
.43032 
.43067 
.43101 


2-32756 
2-32570 
2-32383 
2.32197 
2-32012 


20 

21 
22 
23 
24 


.37999 
.38026 
•38053 
.38080 
.38107 


.92499 
.92488 
.92477 
.92466 
.92455 


.41081 
.41115 
.41149 
.41183 
.41217 


2.43422 
2.43220 
2-43019 
2-42819 
2-42618 


•39608 
•39635 
•39661 
.39688 
.39715 


-91822 
-91810 
-91799 
.91787 
.91775 


.43136 
.43170 
.43205 
.43239 
.43274 


2-31826 
2-31641 
2-31456 
2-31271 
2-31086 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•38134 
•38161 
.38188 
.38215 
.38241 


.92444 
.92432 
.92421 
.92410 
.92399 


.41251 
.41285 
.41319 
.41353 
.41387 

.41421 
.41455 
.41490 
.41524 
.41558 


2.42418 
2.42218 
2.42019 
2.41819 
2.41620 


.39741 
.39768 
.39795 
.39822 
.39848 


-91764 
-91752 
.91741 
.91729 
•91718 


.43308 
.43343 
.43378 
.43412 
.43447 


2.30902 
2.30718 
2.30534 
2.30351 
2.30167 


35 
34 
33 
32 
31 

30 

29 
28 
27 
26 

25 
24 
23 
22 
21 


30 

31 
32 
33 
34 


.38268 
.38295 
.38322 
.38349 
.38376 


.92388 
.92377 
.92366 
.92355 
.92343 


2 41421 
2.41223 
2.41025 
2-40827 
2.40629 


.39875 
•39902 
•39928 
•39955 
.39982 


•91706 
•91694 
•91683 
.91671 
.91660 


.43481 
.43516 
.43550 
.43585 
.43620 


2.29984 
2.29801 
2.29619 
2.29437 
2.29254 


35 
36 
37 
38 
39 


.38403 
.38430 
.38456 
.38483 
•38510 


.92332 
.92321 
.92310 
.92299 
.92287 


.41592 
.41626 
.41660 
.41694 
.41728 


2-40432 
2-40235 
2.40038 
2.39841 
2-39645 


.40008 
.40035 
•40062 
•40088 
.40115 


-91648 
.91636 
.91625 
.91613 
.91601 


.43654 
.43689 
.43724 
.43758 
•43793 


2.29073 
2-28891 
2-28710 
2.28528 
2.28348 


40 

41 
42 
43 
44 


.38537 
.38564 
.38591 
.38617 
.38644 


.92276 
.92265 
.92254 
.92243 
.92231 


.41763 
.41797 
.41831 
.41865 
.41899 


2-39449 
2.39253 
2.39058 
2.38863 
2-38668 


•40141 
•40168 
•40195 
.40221 
.40248 


.91590 
.91578 
.91566 
.91555 
.91543 


.43828 
.43862 
.43897 
.43932 
.43966 


2.28167 
2.27987 
2.27806 
2.27626 
2.27447 


30 

19 
18 
17 
16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 
4 
3 
2 
1 



45 
46 
47 
48 
49 


^38671 
.38698 
.38725 
.38752 
38778 


.92220 
.92209 
.92198 
.92186 
.92175 


.41933 
.41968 
.42002 
.42036 
.42070 


2-38473 
2-38279 
2-38084 
2.37891 
2.37697 


-40275 
40301 
.40328 
.40355 
•40381 


.91531 
.91519 
.91508 
.91496 
.91484 


.44001 
.44036 
.44071 
.44105 
.44140 


2.27267 
2.27088 
2.26909 
2.26730 
2.26552 


50 

51 
52 
53 
54 


.38805 
.38832 
.38859 
.38886 
.38912 


.92164 
.92152 
.92141 
.92130 
.92119 


.42105 
.42139 
.42173 
.42207 
.42242 


2.37504 
2.37311 
2.37118 
2.36925 
2-36733 


•40408 
•40434 
•40461 
•40488 
•40514 


.91472 
.91461 
.91449 
•91437 
•91425 


.44175 
.44210 
.44244 
.44279 
.44314 


2.26374 
2.26196 
2.26018 
2.25840 
2.25663 


55 
56 
57 
58 
59 


•38939 
•38966 
.38993 
.39020 
39046 


.92107 
.92096 
.92085 
.92073 
.92062 


.42276 
.42310 
.42345 
.42379 
.42413 


2-36541 
2-36349 
2-36158 
2-35967 
2.35776 


-40541 
-40567 
•40594 
•40621 
•40647 


•91414 
.91402 
.91390 
.91378 
.91366 


.44349 
.44384 
.44418 
.44453 
•44488 


2^25486 
2^25309 
2^25132 
2^24956 
2^24780 
2.24604 
Tan. 


60 


•39073 


.92050 


.42447 


2.35585 


•40674 


.91355 


•44523 


/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 1 


Cot. 


/ 



67° 



7oa 



66° 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
34^ 35° 



t 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. Cot. 


r 




1 

2 
3 
4 


.40674 
.40700 
.40727 
.40753 
.40780 


.91355 
•91343 
.91331 
.91319 
.91307 


•44523 
•44558 
•44593 
.44627 
.44662 


2.24604 
2.24428 
2.24252 
2.24077 
2.23902 


.42262 
.42288 
.42315 
.42341 
.42367 


•90631 
•90618 
.90606 
•90594 
.90582 


.46631 
.46666 
.46702 
.46737 
.46772 


2.14451 
2.14288 
2.14125 
2.13963 
2.13801 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.40806 
.40833 
.40860 
.40886 
.40913 


.91295 
.91283 
.91272 
.91260 
.91248 


.44697 
.44732 
.44767 
.44802 
.44837 


2.23727 
2.23553 
2.23378 
2.23204 
2.23030 


.42394 
.42420 
.42446 
.42473 
•42499 


•90569 
.90557 
•90545 
.90532 
.90520 


.46808 
.46843 
.46879 
.46914 
.46950 


2.13639 
2.13477 
2.13316 
2.13154 
2.12993 


55 
54 
53 
52 
51 


10 

11 
12 
13 

14 


.40939 
.40966 
.40992 
.41019 
.41045 


.91236 
.91224 
.91212 
.91200 
.91188 


.44872 
.44907 
.44942 
•44977 
.45012 


2.22857 
2.22683 
2.22510 
2.22337 
2.22164 


•42525 
.42552 
.42578 
.42604 
.42631 


.90507 
.90495 
.90483 
-.90470 
•90458 


.46985 
.47021 
.47056 
.47092 
.47128 


2.12832 
2.12671 
2.12511 
2.12350 
2.12190 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


.41072 
.41098 
.41125 
.41151 
.41178 


.91176 
.91164 
.91152 
.91140 
.91128 


.45047 
•45082 
•45117 
•45152 
.45187 


2-21992 
2.21819 
2.21647 
2.21475 
2.21304 


.42657 
.42683 
.42709 
.42736 
.42762 


.90446 
.90433 
.90421 
.90408 
.90396 


.47163 
.47199 
.47234 
•47270 
.47305 


2.12030 
2-11871 
2.11711 
2.11552 
2.11392 


45 
44 
43 
42 
41 


20 

21 
22 
23 
24 


.41204 
.41231 
.41257 
.41284 
.41310 


.91116 
.91104 
.91092 
.91080 
.91068 


•45222 
•45257 
.45292 
.45327 
.45362 


2.21132 
2.20961 
2.20790 
2.20619 
2.20449 


.42788 
.42815 
.42841 
•42867 
.42894 


•90383 
.90371 
.90358 
.90346 
•90334 


•47341 
.47377 
.47412 
.47448 
•47483 


2.11233 
2.11075 
2.10916 
2.10758 
2.10600 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


.41337 
•41363 
.41390 
.41416 
.41443 


•91056 
•91044 
•91032 
.91020 
.91008 


•45397 
.45432 
.45467 
.45502 
•45538 


2.20278 
2.20108 
2.19938 
2.19769 
2.19599 


.42920 
.42946 
.42972 
.42999 
.43025 


•90321 
•90309 
•90296 
.90284 
.90271 


•47519 
•47555 
•47590 
•47626 
•47662 


2.10442 
2-10284 
2-10126 
2-09969 
2.09811 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.41469 
.41496 
•41522 
.41549 
■41575 


.90996 
.90984 
.90972 
.90960 
.90948 
•90936 
•90924 
.90911 
.90899 
•90887 


•45573 
.45608 
.45643 
.45678 
.45713 


2.19430 
2.19261 
2.19092 
2^18923 
2.18755 


.43051 
.43077 
-43104 
.43130 
•43156 


•90259 
•90246 
.90233 
.90221 
.90208 


.47698 
.A7733 
.47769 
•47805 
•47840 


2.09654 
2.09498 
2.09341 
2.09184 
2.09028 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.41602 
.41628 
.41655 
.41681 
.41707 


.45748 
.45784 
.45819 
.45854 
.45889 


2.18587 
2-18419 
2.18251 
2.18084 
2.17916 


.43182 
.43209 
.43235 
.43261 
.43287 


.90196 
.90183 
.90171 
.90158 
•90146 
•90133 
.90120 
.90108 
.90095 
.90082 


•47876 
•47912 
•47948 
•47984 
.48019 

.48055 
.48091 
.48127 
.48163 
.48198 


2.08872 
2.08716 
2.08560 
2.08405 
2.08250 

2.08094 
2.07939 
2-07785 
2.07630 
2.07476 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.41734 
.41760 
.41787 
.41813 
.41840 


•90875 
•90863 
90851 
.90839 
•90826 


.45924 
.45960 
•45995 
•46030 
•46065 


2^17749 
2.17582 
2.17416 
2^17249 
2.17083 


.43313 
.43340 
.43366 
.43392 
•43418 


30 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.41866 
•41892 
•41919 
•41945 
.41972 


•90314 
.90802 
.90790 
.90778 
.90766 


•46101 
•46136 
•46171 
.46206 
.46242 


2.16917 
2^16751 
2^16585 
2.16420 
2.16255 


.43445 
.43471 
.43497 
.43523 
.43549 


.90070 
.90057 
.90045 
.90032 
.90019 


.48234 
•48270 
•48306 
.48342 
.48378 


2.07321 
2.07167 
2-07014 
2.06860 
2.06706 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.41998 
.42024 
42051 
.42077 
.42104 


.90753 
•90741 
.90729 
.90717 
90704 


.46277 
.46312 
.46348 
.46383 
.46418 


2.16090 
2.15925 
2.15760 
2.15596 
2.15432 


.43575 
.43602 
.43628 
.43654 
.43680 


.90007 
.89994 
.89981 
.89968 
.89956 


.48414 
•48450 
•48486 
•48521 
.48557 


2.06553 
2.06400 
2-06247 
2-06094 
2-05942 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


.42130 
.42156 
.42183 
.42209 
.42235 


.90692 
.90680 
.90668 
.90655 
.90643 


.46454 
.46489 
.46525 
.46560 
.46595 


2.15268 
2.15104 
2.14940 
2. 14777 
2.14614 


.43706 
.43733 
-43759 
•43785 
•43811 


.89943 
.89930 
.89918 
.89905 
.89892 


.48593 
•48629 
•48665 
•48701 
.48737 


2-05790 
2.05637 
2.05485 
2.05333 
2.05182 


5 
4 
3 
2 
1 


60 


.42262 


.90631 


.46631 


2^14451 


.43837 


.89879 


.48773 


2.05030 





/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 



65^ 



707 



64° 



-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
26° 37° 





Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 




.43837 
.43863 
.43889 
.43916 
.43942 


.89879 
.89867 
.89854 
.89841 
.89828 


.48773 

.48809 

.48845 

.48881 

.48917_ 

.48953 

.48989 

.49026 

.49062 

.49098 


2.05030 
2.04879 
2.04728 
2.04577 
2.04426 


.45399 
.45425 
.45451 
.45477 
.45503 


.89101 
.89087 
.89074 
.89061 
.89048 


.50953 
.50989 
.51026 
.51063 
.51099 


1.96261 
1.96120 
1.95979 
1.95838 
1.95698 


60 

59 
58 
57 
56 




.43968 
.43994 
.44020 
. 44046 
.44072 


.89816 
.89803 
.89790 
.89777 
.89764 


2.04276 
2.04125 
2.03975 
2.03825 
2.03675 


.45529 
.45554 
.45580 
.45606 
-45632 


.89035 
.89021 
.89008 
.88995 
.88981 


.51136 
.51173 
.51209 
.51246 
.51283 


1.95557 
1.95417 
1.95277 
1.95137 
1.94997 


55 
54 
53 
52 
51 


> 


.44098 
.44124 
.44151 
.44177 
.44203 


.89752 
.89739 
.89726 
.89713 
.89700 


.49134 
.49170 
.49206 
.49242 
.49278 
.49315 
.49351 
.49387 
.49423 
.49459 


2.03526 
2.03376 
2.03227 
'2.03078 
2.02929 


.45658 
.45684 
.45710 
.45736 
.45762 


.88968 
.88955 
.88942 
.88928 
.88915 


.51319 
.51356 
.51393 
.51430 
.51467 


1.94858 
1.94718 
1.94579 
1.94440 
1.94301 


50 

49 
48 
47 
46 




.44229 
.44255 
.44281 
.44307 
.44333 


.89687 
.89674 
.89662 
.89649 
.89636 


2.02780 
2.02631 
2.02483 
2.02335 
2.02187 


.45787 
.45813 
.45839 
.45865 
.45891 


.88902 
.88888 
.88875 
.88862 
.88848 


.51503 
.51540 
.51577 
.51614 
.51651 


1.94162 
1.94023 
1.93885 
1.93746 
1.93608 


45 
44 
43 
42 
41 


) 


.44359 
.44385 
.44411 
.44437 
.44464 


.89623 
.89610 
.89597 
.89584 
.89571 


.49495 
.49532 
.49568 
.49604 
.49640 


2.02039 
2.01891 
2.01743 
2.01596 
2.01449 


.45917 
.45942 
.45968 
■45994 
.46020 


.88835 
.88822 
.88808 
-88795 
.88782 


.51688 
.51724 
.51761 
.51798 
.51835 


1.93470 
1.93332 
1.93195 
1.93057 
1.92920 


40 

39 
38 
37 
36 




.44490 
.44516 
.44542 
.44568 
.44594 


.89558 
.89545 
.89532 
.89519 
.89506 


.49677 
.49713 
.49749 
.49786 
.49822 


2.01302 
2.01155 
2.01008 
2.00862 
2.00715 


.46046 
.46072 
.46097 
.46123 
•46149 


.88768 
.88755 
.88741 
.88728 
.88715 


.51872 
.51909 
.51946 
.51983 
.52020 


1.92782 
1.92645 
1.92508 
1.92371 
1.92235 


35 
34 
33 
32 
31 


> 


.44620 
.44646 
.44672 
•44698 
•44724 
.44750 
•44776 
•44802 
.44828 
• 44854 


.89493 
•89480 
.89467 
.89454 
.89441 
.89428 
.89415 
.89402 
.89389 
.89376 


.49858 
.49894 
.49931 
.49967 
.50004 


2.00569 
2-00423 
2.00277 
2.00131 
1.99986 


.46175 
.46201 
.46226 
.46252 
.46278 


.88701 
.88688 
.88674 
.88661 
.88647 


.52057 
.52094 
.52131 
.52168 
.52205 


1.92098 
1.91962 
1.91826 
1.91690 
1.91554 


30 

29 
28 
27 
26 




. 50040 
.50076 
.50113 
.50149 
.50185 


1.99841 
1.99695 
1.99550 
1.99406 
1.99261 


.46304 
.46330 
.46355 
.46381 
.46407 


.88634 
.88620 
.88607 
.88593 
.88580 


.52242 
.52279 
.52316 
.52353 
.52390 


1.91418 
1.91282 
1.91147 
1.91012 
1.90876 


25 
24 
23 
22 
21 


^ 


•44880 
.44906 
•44932 
.44958 
.44984 


.89363 
.89350 
.89337 
.89324 
.89311 


.50222 
.50258 
.50295 
.50331 
.50368 


1.99116 
1.98972 
1.98828 
1.98684 
1.98540 


.46433 
.46458 
.46484 
.46510 
.46536 


.88566 
.88553 
.88539 
.88526 
.88512 


.52427 
.52464 
.52501 
.52538 
.52575 


1.90741 
1.90607 
1.90472 
1.90637 
1.90203 


30 

19 
18 
17 
16 




.45010 
.45036 
.45062 
.45088 
.45114 


.89298 
.89285 
.89272 
.89259 
.89245 
.89232 
.89219 
.89206 
.89193 
.89180 


. 50404 
. 50441 
.50477 
.50514 
.50550 


1.98396 
1.98253 
1.98110 
1.97966 
1.97823 


.46561 
.46587 
.46613 
.46639 
.46664 


.88499 
.88485 
.88472 
.88458 
.88445 


.52613 
.52650 
.52687 
.52724 
•52761 


1.90069 
1.89935 
1.89801 
1.89667 
1.89533 


15 

14 
13 
12 
11 


> 


.45140 
.45166 
.45192 
.45218 
•45243 


.50587 
.50623 

50660 
.50696 
.50733 
.50769 
.50806 
.50843 
.50879 

50916 


1.97681 
1.97538 
1.97395 
1.97253 
1.97111 


.46690 
.46716 
.46742 
.46767 
.46793 


.88431 
.88417 
.88404 
.88390 
.88377 
.88363 
.88349 
.88336 
.88322 
.88308 


.52798 
.52836 
.52873 
.52910 
.52947 


1.89400 
1.89266 
1.89133 
1.89000 
1.88867 


10 

9 
8 
7 
6 




•45269 
•45295 
•45321 
-45347 
•45373 


.89167 
.89153 
•89140 
•89127 
•89114 


1.96969 
1.96827 
1.96685 
1.96544 
1.96402 


.46819 
.46844 
•46870 
.46896 
•46921 


.52985 
.53022 
.53059 
.53096 
.53134 


1.88734 
1.88602 
1.88469 
1.88337 
1.88205 


5 

4 
3 

2 

1 


> 


•45399 


•89101 


-50953 


1.96261 


•46947 


.88295 


•53171 


1.88073 


.. 




Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 





63^ 



708 



ea^ 



TABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
38° 39° 


"o" 

1 
2 
3 

5 
6 
7 
8 
9 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


f 


.46947 
.46973 
-46999 
.47024 
.47050 


-88295 
-88281 
-88267 
.88254 
.88240 


.53171 
-53208 
53246 
.53283 
-53320 


1.88073 
1-87941 
1-87809 
1-87677 
1-87546 


.48481 
-48506 
-48532 
-48557 
-48583 


87462 
-87448 
-87434 

^7420 
.87406 


-55431 
-55469 
.55507 
.55545 
.55583 


I 80405 
1-80281 
1.80158 
1.80034 
1-79911 


60 

59 
58 
57 
56 


.47076 
.47101 
-47127 
.47153 
.47178 


■88226 
88213 
-88199 
.88185 
•88172 


-53358 
.53395 
.53432 
53470 
-53507 


1-87415 
1-87283 
1-87152 
1.87021 
1-86891 


-48608 
-48634 
.48659 
48684 
■48710 


-87391 
-87377 
.87363 
.87349 
.87335 


.55621 
-55659 
-55697 
■55736 
.55774 


1 79788 
1 79665 
1.79542 
1.79419 
1-79296 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 

15 
16 
17 
18 
19. 


.47204 
.47229 
.47255 
.47281 
-47306 


.88158 
-88144 
-88130 
.88117 
-88103 


-53545 
.53582 
.53620 
.53657 
-53694 


1.86760 
1.86630 
1.86499 
1.86369 
1.86239 


-48735 
-48761 
.48786 
.48811 
-48837 


.87321 
.87306 
.87292 
.87278 
.87264 


.55812 
-55850 
-55888 
-55926 
-55964 


1-79174 
1.7905] 
1.78929 
1-78807 
1-78685 

1.78563 
1-78441 
1-78319 
1-78198 
1-78077 
1.77955 
1-77834 
1-77713 
1-77592 
1-77471 


50 

49 
48 
47 
46 


-47332 
-47358 
-47383 
-47409 
.47434 


.88089 
.88075 
.88062 
.88048 
.88034 


.53732 
.53769 
.53807 
-53844 
.53882 


1.86109 
1.85979 
1.85850 
1-85720 
1-85591 
1-85462 
1-85333 
1-85204 
1-85075 
1-84946 


.48862 
.48888 
.48913 
.48938 
-48964 

-48989 
-49014 
-49040 
-49065 
-49090 


.87250 
.87235 
•87221 
.87207 
.87193 

-87178 
-87164 
-87150 
-87136 
-87121 


-56003 
-56041 
-56079 
-56117 
-56156 
-56194 
-56232 
-56270 
-56309 
-56347 


45 
44 
43 
42 
41 


30 

21 
22 
I 23 
i24 

'25 

;26 

27 

28 

'29 


-47460 
-47486 
.47511 
-47537 
.47562 


.88020 
•88006 
-87993 
-87979 
-87965 


-53920 
.53957 
.53995 
.54032 
.54070 


40 

39 
38 
37 
36 


-47588 
.47614 
47639 
.47665 
.47690 


-87951 
-87937 
-87923 
-87909 
.87896 


.54107 
-54145 
.54183 
-54220 
•54258 

-54296 
-54333 
-54371 
- 54409 
. 54446 


1.84818 
1-84689 
1-84561 
1-84433 
1-84305 


-49116 
-49141 
-49166 
-49192 
•49217 


-87107 
87093 
-87079 
-87064 
-87050 


-56385 
-56424 
-56462 
-56501 
.56539 


1-77351 
1-77230 
1.77110 
1-76990 
1-76869 


35 
34 
33 
32 
31 


,30 

31 
(32 

33 

34 

35 
'36 

37 

38 
i39 

,40 

41 
142 
143 

44 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

60_ 

/ 


.47716 
.47741 
.47767 
.47793 
.47818 


-87882 
-87868 
-87854 
-87840 
-87826 


1-84177 
1-84049 
1-83922 
1-837&4 
1.83667 


-49242 
-49268 
-49293 
.49318 
.49344 


-87036 
-87021 
.87007 
.86993 
-86978 


-56577 
-56616 
.56654 
.56693 
-56731 


1-76749 
1.76629 
1-76510 
1-76390 
1-76271 


30 

29 
28 
27 
26 


-47844 
-47869 
.47895 
.47920 
.47946 
.47971 
-47997 
-48022 
-48048 
-48073 


-87812 
-87798 
-87784 
-87770 
.87756 
•87743 
-87729 
-87715 
-87701 
.87687 


- 54484 
.54522 
-54560 
-54597 
-54635 

-54673 
-54711 

- 54748 
-54786 
.54824 


1.83540 
1-83413 
1-83286 
1-83159 
1-83033 
1-82906 
1-82780 
1-82654 
1-82528 
1-82402 


-49369 
-49394 
-49419 
-49445 
.49470 
-49495 
-49521 
-49546 
-49571 
•49596 


-86964 
-86949 
-86935 
-86921 
.86906 
-86892 
-86878 
-86863 
.86849 
-86834 


-56769 
.56808 
.56846 
-56885 
-56923 
-56962 
.57000 
.57039 
.57078 
-57116 


1-76151 
1-76032 
1-75913 
1-75794 
1-75675 
1-75556 
1.75437 
1-75319 
1.75200 
1-75082 


25 
24 
23 
22 

30 

19 
18 
17 
16 


-48099 
-48124 
-48150 
-48175 
.48201 


-87673 
-87659 
-87645 
.87631 
.87617 


-54862 
-54900 
.54938 
.54975 
.55013 


1-82276 
1-82150 
1-82025 
1-81899 
1-81774 


-49622 
-49647 
.49672 
-49697 
-49723 


.86820 
-86805 
-86791 
-86777 
-86762 


-57155 
-57193 
.57232 
.57271 
-57309 


1-74964 
1-74846 
1-74728 
1-74610 
1-74492 


15 
14 
13 
12 
11 


.48226 
-48252 
-48277 
-48303 
.48328 


-87603 
-87589 
-87575 
-87561 
-87546 


.55051 
.55089 
.55127 
-55165 
.55203 


1-81649 
1-81524 
1-81399 
1-81274 
1-81150 


-49748 
.49773 
-49798 
-49824 
-49849 


-86748 
-86733 
-86719 
-86704 
•86690 


-57348 
.57386 
.57425 
-57464 
.57503 


1-74375 
1-74257 
1 - 74140 
1-74022 
1-73905 


10 

9 
8 
7 
6 


-48354 
-48379 
-48405 
-48430 
-48456 


-87532 
.87518 
.87504 
.87490 
.87476 


.55241 
.55279 
-55317 
-55355 
-55393 


1.81025 
1.80901 
1-80777 
1-80653 
1-80529 


-49874 
-49899 
-49924 
-49950 
.49975 


-86675 
-86661 
-86646 
-86632 
-86617 


-57541 
-57580 
-57619 
-57657 
-57696 


1-73788 
1-73671 
1-73555 
1-73438 
1-73321 


5 

4 
3 

2 

1 


-48481 


.87462 


-55431 


1.80405 


-50000 


-86603 


-57735 


1-73205 





Cos. 


Sin. 


Cot. 


Tan. 


Cos= 


Sin. 


Cot. 


Tan. 


' 






61 


L° 


70 


9 


6 


0° 







TABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
30"* 31° 


/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 







1 
2 
3 
4 


.50000 
.50025 
.50050 
.50076 
.50101 


•86603 
.86588 
•86573 
.86559 
.86544 


•57735 
•57774 
.57813 
.57851 
•57890 


1.73205 
1.73089 
1.72973 
1.72857 
1.72741 


.51504 
.51529 
-51554 
-51579 
-51604 


-85717 
-85702 
-85687 
.85672 
-85657 


-60086 
-60126 
-60165 
. 60205 
.60245 


1-66428 
1.66318 
1.66209 
1.66099 
1-65990 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.50126 
.50151 
.50176 
.50201 
.50227 


.86530 
.86515 
.86501 
•86486 
•86471 


.57929 
.57968 
.58007 
•58046 
.58085 


1.72625 
1.72509 
1.72393 
1^72278 
1^72163 


-51628 
.51653 
-51678 
-51703 
-51728 


-85642 
-85627 
.85612 
.85597 
.85582 


.60284 
.60324 
-60364 

- 60403 

- 60443 


1-65881 
1-65772 
1-65663 
1. 65554 
1.65445 


55 
54 
53 
52 
51 


10 

11 
12 
13 

14 


.50252 
.50277 
.50302 
.50327 
.50352 


•86457 
•86442 
•86427 
•86413 
.86398 


.58124 
.58162 
•58201 
•58240 
.58279 


1-72047 
1-71932 
1^71817 
1.71702 
1.71588 


-51753 
-51778 
-51803 
-51828 
-51852 


.85567 
.85551 
.85536 
.85521 
.85506 


.60483 
-60522 
-60562 
-60602 
.60642 


1-65337 
1-65228 
1-65120 
1-65011 
1.64903 


50 

49 
48 
47 
43 


15 
16 
17 
18 
19 


.50377 
. 50403 
.50428 
.50453 
•50478 


.86384 
.86369 
.86354 
•86340 
•86325 


•58318 
•58357 
•58396 
•58435 
•58474 


1-71473 
1.71358 
1.71244 
1.71129 
1.71015 


.51877 
.51902 
.51927 
.51952 
.51977 


.85491 
.85476 
-85461 
•85446 
•85431 


-60681 
.60721 
.60761 
.60801 
.60841 


1-64795 
1.64687 
1.64579 
1.64471 
1-64363 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


.50503 
.50528 
.50553 
.50578 
.50603 


.86310 
.86295 
•86281 
•86266 
.86251 , 


•58513 
•58552 
•58591 
.58631 
.58670 


1-70901 
1-70787 
1.70673 
1.70560 
1.70446 


.52002 
=52026 
.52051 
.52076 
.52101 


•85416 
.85401 
.85385 
•85370 
.85355 


-60881 
-60921 
-60960 
-61000 
•61040 


1-64256 
1-64148 
1-64041 
1-63934 
1-63826 


40 

39 
38 
37 
36 


25 
26 
27 
28 
?9 


.50628 
.50654 
.50679 
.50704 
.50729 


.86237 
.86222 
.86207 
.86192 
•86178 


58709 
.58748 
.58787 
.58826 
.58865 


1.70332 
1.70219 
1.70106 
1.69992 
1-69879 


.52126 
.52151 
.52175 
.52200 
.52225 


.85340 
.85325 
•85310 
•85294 
.85279 


.61080 
.61120 
.61160 
.61200 
.61240 


1.63719 
1.63612 
1.63505 
1.63398 
1.63292 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


.50754 

.50779 

. 50804 

50829 

50854 


.86163 
.86148 
.86133 
.86119 
.86104 


.58905 
.58944 
.58983 
.59022 
.59061 


1-69766 
1.69653 
1.69541 
1.69428 
1.69316 


.52250 
.52275 
-52299 
.52324 
.52349 


.85264 
.85249 
.85234 
.85218 
-85203 


.61280 
.61320 
.61360 
.61400 
.61440 


1.63185 
1.63079 
1.62972 
1.62866 
1-62760 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.50879 
.50904 
.50929 
.50954 
.50979 


.86089 
.86074 
.86059 
•86045 
•86030 


.59101 
•59140 
•59179 
59218 
.59258 


1.69203 
1-69091 
1-68979 
1-68866 
1.68754 


•52374 
•52399 
•52423 
. 52448 
•52473 


-85188 
-85173 
.85157 
.85142 
.85127 


.61480 
.61520 
.61561 
.61601 
-61641 


1.62654 
1.62548 
1.62442 
1.62336 
1-62230 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


.51004 
.51029 
.51054 
.51079 
51104 


•86015 
•86000 
•85985 
.85970 
.85956 


.59297 
.59336 
.59376 
.59415 
.59454 


1.68643 
1.68531 
1.68419 
1.68308 
1.68196 


■52498 
.52522 
-52547 
•52572 
•52597 


.85112 
.85096 
.85081 
.85066 
.85051 


.61681 
.61721 
.61761 
.61801 
.61842 


1.62125 
3.62019 
1.61914 
1.61808 
1.61703 


20 

19 
18 
17 
16 


45 
46 
47 
48 
49 


.51129 
.51154 
•51179 
.51204 
.51229 


.85941 
.85926 
.85911 
.85896 
•85881 


.59494 
.59533 
.59573 
•59612 
•59651 


1.68085 
1.67974 
1.67863 
1-67752 
1.67641 


•52621 
•52646 
•52671 
•52696 
•52720 


-85035 
-85020 
-85005 
.84989 
•84974 


.61882 
-61922 
.61962 
.62003 
. 62043 


1.61598 
1.61493 
1.61388 
1.61283 
1.61179 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 


.51254 
•51279 
.51304 
•51329 
•51354 


.85866 
.85851 
•85836 
•85821 
•85806 


•59691 
•59730 
•59770 
.59809 
•59849 


1.67530 
1.67419 
1.67309 
1.67198 
1-67088 
1.66978 
1.66867 
1.66757 
1.66647 
1-66538 


•52745 
-52770 
-52794 
•52319 
•52844 


.84959 
. 84943 
.84928 
.84913 
•84897 


-62083 
-62124 
.62164 
.62204 
.62245 


1.61074 
1.60970 
1-60865 
1.60761 
1.60657 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•51379 
51404 
.51429 
•51454 
•51479 
•51504 


•85792 
•85777 
•85762 
•85747 
•85732 


•59888 
•59928 
•59967 
•60007 
.60046 


•52869 
-52893 
-52918 
•52943 
-52967 


•84882 
.84866 
-84851 
-84836 
.84820 


.62285 
.62325 
.62366 
.62406 
.62446 


1.60553 
1.60449 
1.60345 
1 . 60241 
1.60137 


5 
4 
3 
2 
1 


§2. 


85717 


•60086 


1-66428 


-52992 
Cos. 


.84805 
Sin. 


-62487 
Cot. 


1 . 60033 
tTan. 


-_o 


/ 


Cos. 1 Sin. 


Cot. 


Tan. 


/ ■ 



59' 



no 



68" 



^ TABLE IX.- 


-NATURAL SINES. COSINES, TANGENTS. AND COTANGENTS 


1?" 

3 
8 4 

1 5 
7 

2 8 
1 9 

) 10 
11 
12 
13 
14 

15 
16 
17 
18 
19 
30 
21 
22 

,24 


Sin. 


Cos 


Tan. 


Cot 


Sin. 


Cos. 


Tan. 


Cot. 


# 


.52992 
.53017 
.53041 
.53066 
.53091 


.84805 
.84789 
.84774 
.84759 
.84743 


.62487 
.62527 
.62568 
.62608 
•62649 


1.60033 
1.59930 
1.59826 
1.59723 
1.59620 


.54464 
•54488 
•54513 
•54537 
.54561 


•83867 
.83851 
•83835 
.83819 
.83804 


.64941 
.64982 
.65024 
.65065 
•65106 


1.53986 
1.53888 
1.53791 
1.53693 
1.53595 


60 

59 
58 
57 
56 


.53115 
.53140 
.53164 
.53189 
.53214 


.84728 
.84712 
.84697 
.84681 
.84666 


.62689 
.62730 
.62770 
.62811 
.62852 


1.59517 
1.59414 
1.59311 
1.59208 
1.59105 


.54585 
.54610 
•54635 
.54659 
•54683 


•83788 
.83772 
.83756 
.83740 
.83724 


.65148 
•65189 
•65231 
.65272 
.65314 


1-53497 
1-53400 
1.53302 
1.53205 
1-53107 


55 
54 
53 
52 
51 


.53238 
.53263 
.53288 
.53312 
.53337 


.84650 
.84635 
.84619 
.84604 
.84588 


.62892 
.62933 
.62973 
.63014 
.63055 


1.59002 
1.58900 
1.58797 
1.58695 
1.58593 


•54708 
-54732 
•54756 
•54781 
•54805 


.83708 
.83692 
.83676 
.83660 
•83645 


.65355 
.65397 
.65438 
.65480 
•65521 


1.53010 
1.52913 
1.52816 
1.52719 
1.52622 


50 

49 
48 
47 
46 


.53361 
.53386 
.53411 
•53435 
.53460 


.84573 
.84557 
.84542 
.84526 
.84511 


.63095 
.63136 
.63177 
.63217 
•63258 


1.58490 
1.58388 
1.58286 
1.58184 
1.58083 


•54829 
.54854 
•54878 
.54902 
•54927 


.83629 
.83613 
.83597 
.83581 
•83565 


.65563 
.65604 
.65646 
.65688 
•65729 


1.52525 
1.52429 
1.52332 
1.52235 
1-52139 


45 
44 
43 
42 
41 


•53484 
.53509 
.53534 
.53558 
•53583 


.84495 
.84480 
.84464 
•84448 
.84433 


.63299 
.63340 
.63380 
.63421 
•63462 


1.57981 
1.57879 
1.57778 
1.57676 
1.57575 


■54951 
•54975 
•54999 
•55024 
•55048 


.83549 
.83533 
.83517 
.83501 
.83485 


.65771 
.65813 
.65854 
.65896 
.65938 


1.52043 
1.51946 
1.51850 
1.51754 
1.51658 


40 

39 
38 
37 

36 


25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 


.53607 
.53632 
.53656 
•53681 
.53705 

.53730 
.53754 
.53779 
.53804 
.53828 
.53853 
.53877 
.53902 
.53926 
.53951 


•84417 
.84402 
.84386 
•84370 
.84355 


•63503 
.63544 
•63584 
•63625 
•63666 


1.57474 
1.57372 
1.57271 
1.57170 
1.57069 


•55072 
•55097 
•55121 
•55145 
•55169 


.83469 
.83453 
.83437 
•83421 
•83405 


.65980 
•66021 
.66063 
.66105 
.66147 


1.51562 
1.51466 
1.51370 
1-51275 
1-51179 


35 
34 
33 
32 
31 


.84339 

.84324 
.84308 
•84292 
•84277 


•63707 
•63748 
•63789 
.63830 
•63871 


1.56969 
1.56868 
1.56767 
1.56667 
1.56566 


.55194 
.55218 
.55242 
.55266 
•55291 
•55315 
•55339 
.55363 
•55388 
.55412 


•83389 
.83373 
• 8335.6 
.83340 
•83324 


.66189 
.66230 
.66272 
.66314 
.66356 


1.51084 
1.50988 
1.50893 
1-50797 
1-50702 


30 

29 
28 
27 
26 


•84261 
.84245 
.84230 
.84214 
.84198 


.63912 
.63953 
.63994 
.64035 
•64076 


1.56466 
1.56366 
1.56265 
1.56165 
1.56065 


.83308 
.83292 
.83276 
•83260 
•83244 


.66398 
.66440 
.66482 
.66524 
.66566 


1-50607 
1.50512 
1-50417 
1.50322 
1-50228 


25 
24 
23 
22 
21 


40 

141 

!42 
43 
144 

45 
46 
47 
'48 
,49 

150 

51 

^52 

53 

54 


•53975 
.54000 
.54024 
.54049 
.54073 


•84182 
•84167 
•84151 
•84135 
.84120 
.84104 
•84088 
•84072 
•84057 
•84041 


.64117 
.64158 
.64199 
. 64240 
•64281 


1.55966 
1.55866 
1.55766 
1.55666 
1.55567 


•55436 
•55460 
.55484 
.55509 
•55533 


•83228 
•83212 
.33195 
•83179 
•83163 


.66608 
.66650 
.66692 
.66734 
.66776 


1.50133 
1.50038 
1.49944 
1.49849 
1-49755 


30 

19 
18 
17 
16 


.54097 
•54122 
.54146 
.54171 
•54195 


.64322 
.64363 
. 64404 
. 64446 
•64487 


1.55467 
1.55368 
1.55269 
1.55170 
1.55071 


•55557 
•55581 
•55605 
•55630 
•55654 


.83147 
.83131 
.83115 
.83098 
•83082 


.66818 
.66860 
.66902 
.66944 
.66986 


1.49661 
1.49566 
1.49472 
1-49378 
1-49284 


15 
14 
13 
12 
11 


.54220 
• 54244 
.54269 
.54293 
.54317 


•84025 
•84009 
•83994 
•83978 
•83962 


.64528 
.64569 
•64610 
•64652 
•64693 


1.54972 
1.54873 
1.54774 
1.54675 
1.54576 


•55678 
•55702 
•55726 
•55750 
•55775 


•83066 
•83050 
•83034 
.83017 
83001 


.67028 
.67071 
.67113 
.67155 
.67197 


1.49190 
1.49097 
1.49003 
1.48909 
1-48816 


10 

9 
8 
7 

6 


55 

56 

57 

<58 

;59 


.54342 
-54366 
•54391 
•54415 
• 54440 


.83946 
•83930 
•83915 
•83899 
•83883 


•64734 
•64775 
•64817 
.64858 
64899 


1.54478 
1.54379 
1.54281 
1.54183 
1.54085 


•55799 
•55823 
•55847 
•55871 
•55895 


•82985 
82969 
•82953 
•82936 
•82920 


.67239 
.67282 
.67324 
.67366 
.67409 


1.48722 
1.48629 
1-48536 
1-48442 
1.48349 


5 

4 
3 

2 

1 


60 


• 54464 


•83867 


-64941 


1. 53986 


•55919 


•82904 


.67451 


1.48256 


O 


\ / 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


■ tan. 


' 



57** 



711 



56« 



TABLE IX.- 


-NATURAL S NES, COSINES, TANGENTS, AND COTANGENTS. 
34° 35° 




Sin. 


Cos. 


Tan. 


Cot. 


Sin, 


Cos. 


Tan. 


Cot. 


f 




1 
2 
3 

4 


.55919 
•55943 
•55968 
•55992 
.56016 


•82904 
.82887 
.82871 
.82855 
82839 


•67451 
.67493 
.67536 
.67578 
•67620 


1.48256 
1.48163 
1.48070 
1.47977 
1.47885 


•57358 
•57381 
•57405 
•57429 
.57453 


•81915 
.81899 
.81882 
.81865 
.81848 


.70021 
.70064 
•70107 
.70151 
.70194 


1.42815 
1.42726 
1.42638 
1.42550 
1.42462 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 


5 
6 
7 
8 
9 


•56040 
•56064 
.56088 
.56112 
.56136 


.82822 
.82806 
.82790 
.82773 
.82757 


.67663 
.67705 
.67748 
.67790 
.67832 


1.47792 
1.47699 
1.47607 
1.47514 
1.47422 


.57477 
.57501 
.57524 
.57548 
.57572 


.81832 
.81815 
.81798 
.81782 
.81765 


•70238 
.70281 
.70325 
.70368 
.70412 


1.42374 
1.42286 
1.42198 
1.42110 
1.42022 


10 

11 
12 
13 

14 


.56160 
.56184 
.56208 
.56232 
.56256 


.82741 
.82724 
.82708 
.82692 
.82675 


.67875 
.67917 
.67960 
.68002 
•68045 


1.47330 
1.47238 
1.47146 
1.47053 
1.46962 


.57596 
.57619 
.57643 
•57667 
•57691 


.81748 
.81731 
.81714 
.81698 
-81681 


.70455 
.70499 
. 70542 
.70586 
.70629 


1.4.1934 
1.41847 
1.41759 
1.41672 
1.41584 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19^ 


.56280 
.56305 
.56329 
.56353 
.56377 


.82659 
.82643 
.82626 
.82610 
.82593 


.68088 
.68130 
.68173 
.68215 
.68258 


1.46870 
1.46778 
1.46686 
1.46595 
1.46503 


•57715 
•57738 
•57762 
•57786 
•57810 


.81664 
.81647 
.81631 
.81614 
.81597 


.70673 
.70717 
.70760 
.70804 
. 70848 


1.41497 
1.41409 
1.41322 
1.41235 
1.41148 


45 
44 
43 
42 
41 


»0 

21 
22 
23 
24 


•56401 
.56425 
•56449 
•56473 
•56497 


.82577 
.82561 
.82544 
.82528 
•82511 


.68301 
.68343 
.68386 
.68429 
•68471 


1.46411 
1.46320 
1.46229 
1.46137 
1.46046 


•57833 
•57857 
•57881 
•57904 
.57928 


.81580 
.81563 
.81546 
.81530 
.81513 


.70891 
.70935 
.70979 
•71023 
•71066 


1.41061 
1.40974 
1.40887 
1.40800 
1.40714 


40 

39 
38 
37 
36 
35 
34 
33 
32 
31 


25 
26 
27 
28 
29 


•56521 
•56545 
.56569 
.56593 
•56617 


.82495 
.82478 
.82462 
.82446 
•82429 


.68514 
.68557 
.68600 
.68642 
.68685 


1.45955 
1.45864 
1.45773 
1.45682 
1.45592 


•57952 
•57976 
•57999 
•58023 
.58047 


.81496 
•81479 
.81462 
.81445 
.81428 


•71110 
•71154 
.71198 
.71242 
.71285 


1.40627 
1.40540 
1.40454 
1.40367 
1^40281 


30 

31 
32 
33 
34 


.56641 
•56665 
•56689 
•56713 
•56736 


.82413 
.82396 
.82380 
.82363 
.82347 


.68728 
.68771 
.68814 
.68857 
.68900 


1.45501 
1.45410 
1.45320 
1.45229 
1.45139 


.58070 
.58094 
.58118 
.58141 
.58165 


.81412 
.81395 
.81378 
.81361 
.81344 


•71329 
•71373 
.71417 
.71461 
•71505 


1.40195 
1.40109 
1.40022 
1.39936 
1.39850 


30 

29 
28 
27 
26 
25 
24 
23 
22 
21 


35 
36 
37 
38 
39 


•56760 
•56784 
•56803 
•56832 
• 568fi6_ 
•56880 
.56904 
.56928 
.56952 
.56976 


.82330 
.82314 
.82297 
.82281 
.82264 

.82248 
.82231 
•82214 
•82198 
.82181 


.68942 
-68985 
.69028 
.69071 
.69114 

.69157 
.69200 
.69243 
.69286 
•69329 


1.45049 
1.44958 
1.44868 
1-44778 
1.44688 


•58189 
.58212 
.582.36 
.58260 
•58283 


.81327 
.81310 
.81293 
.81276 
.81259 


•71549 
•71593 
• 7163.7 
.71681 
.71725 


1.39764 
1.39679 
1.39503 
1.39507 
1.39421 


40 

41 
42 
43 
44 


1.44598 
1.44508 
1.44418 
1.44329 
1.44239 


•58307 
•58330 
•58354 
•58378 
•58401 


.81242 
.81225 
.81208 
.81191 
.81174 


.71769 
.71813 
.71857 
.71901 
.71946 


1.39336 
1.39250 
1.39165 
1.39079 
1.38994 


30 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 



f 


45 

46 
47 
48 
49 


.57000 
.57024 
.57047 
.57071 
.57095 


.82165 
•82148 
•82132 
•82115 
•82098 


•69372 
•69416 
•69459 
•69502 
•69545 


1.44149 
1.44060 
1.43970 
1.43881 
1.43792 
1.43703 
1.43614 
1.43525 
1.43436 
1.43347 


•58425 
. 58449 
•58472 
-58496 
.58519 


.81157 
.81140 
•81123 
81106 
•81089 


-71990 
.72034 
.72078 
.72122 
.72167 


1.38909 
1^ 38824 
1^38738 
1.38653 
1.38568 


50 

51 
52 
53 
54 


.57119 
.57143 
•57167 
•57191 
.57215 


•82082 
•82065 
•82048 
.82032 
•82015 


•69588 
•69631 
•69675 
•69718 
.69761 


.58543 
•58567 
•58590 
•58614 
-58637 


•81072 
•81055 
•81038 
•81021 
•81004 


.72211 
.72255 
.72299 
.72344 
.72388 


1.38484 
1.38399 
1.38314 
1.38229 
1.38145 


55 
56 
57 
58 
59 


•57238 
•57262 
•57286 
•57310 
•57334 
•57358 


•81999 
•81982 
81965 
81949 
81932 


.69804 
.69847 
•69891 
.69934 
.69977 


1.43258 
1.43169 
!• 43080 
1.42992 
1.42903 


58661 
■58684 
•58708 
•58731 
•58755 


•80987 
•80970 
•80953 
•80936 
•80919 


.72432 
.72477 
.72521 
.72565 
.72610 


1.38060 
1.37976 
1.37891 
1.37807 
1.37722 

1.37638 


60 


81915 


.70021 


1.42815 


•58779 


.80902 


•72654 


"^ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 



66^ 



712 



54^ 



TABLE IX.— NATURAL SINES, COSINES, TANGENTS, AND COTANGENTa 
36"* 37° 





Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


r 




1 

2 
3 

4 

5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 


58779 
•58802 
-58826 
•58849 
•58873 


.80902 
.80885 
•80867 
.80850 
•80833 
•80816 
.80799 
.80782 
.80765 
•80748 


.72654 
•72699 
.72743 
.72788 
.72832 


1.37638 
1.37554 
1.37470 
1^ 37386 
1^37302 


•60182 
60205 
•60228 
•60251 
. 60274 


79864 

79846 

79829 

-79811 

-79793 


75355' 

75401 

75447 

75492 

75538 


1.32704 
1.32624 
1-32544 
1.32464 
1.32384 


60 

59 
58 
57 
56 


58896 
•58920 
•58943 
•58967 

58990 


.72877 
.72921 
.72966 
.73010 
•73055 
•73100 
.73144 
•73189 
•73234 
•73278 


1-37218 
1.37134 
1-37050 
1.36967 
1-36883 


.60298 1 

.60321 

-60344 

-60367 

-60390 


-79776 
-79758 
-79741 
-79723 
-79706 


•75584 
•75629 
-75675 
-75721 
-75767 


1.32304 
1.32224 
1.32144 
1.32064 
1.31984 


55 
54 
53 
52 
51 


•59014 
•59037 
•59061 
. 59084 
•59108 


.80730 
.80713 
.80696 
.80679 
.80662 


1.36800 
1.36716 
1.36633 
1.36549 
1.36466 


• 60414 
-60437 

• 60460 
.60483 
.60506 


.79688 
.79671 
.79653 
.79635 
•79618 


-75812 
.75858 
.75904 
-75950 
.75996 


1-31904 
1-31825 
1-31745 
1-31666 
1-31586 


50 

49 
48 
47 
46 


•59131 
•59154 
.59178 
.59201 

.59225 


.80644 
•80627 
.80610 
.80593 
•80576 


•73323 
.73368 
•73413 
.73457 
.73502 


1-36383 
1.36300 
1.36217 
1.36134 
1.36051 


.60529 
.60553 
.60576 
•60599 
.60622 


•79600 
.79583 
•79565 
-79547 
-79530 


.76042 
-76088 
.76134 
-76180 
76226 


1-31507 
1-31427 
1-31348 
1-31269 
1-31190 


45 
44 
43 
42 
41 


20 

21 
,22 

|23 
.24 


•59248 
•59272 
•59295 
•59318 
•59342 


•80558 
•80541 
•80524 
.80507 
•80489 


•73547 
•73592 
•73637 
•73681 
•73726 

•73771 
•73816 
•73861 
•73906 
•73951 


1.35968 
1.35885 
1.35802 
1.35719 
1.35637 


.60645 
.60668 
.60691 
.60714 
.60738 


■79512 
.79494 
.79477 
.79459 
• 79441 


-76272 
•76318 
-76364 
.76410 
-76456 


1-31110 
1-31031 
1.30952 
1.30873 
1.30795 


40 

39 
38 
37 
36 


i25 
'26 

27 
128 

29 

30 
;31 

32 
'33 

34 


.59365 
•59389 
.59412 
•59436 
•59459 


•80472 
- 80455 
•80438 
•80420 
• 80403 


1.35554 
1.35472 
1.35389 
1.35307 
1.35224 


. 60761 
•60784 
•60807 
•60830 
•60853 


.79424 
. 79406 
•79388 
.79371 
•79353 


-76502 
-76548 
.76594 
.76640 
-76686 


1-30716 
1^30637 
1^30558 
1-30480 
1-30401 


35 
34 
33 
32 
31 


.59482 
•59506 
•59529 
.59552 
.59576 


•80386 
•80368 
•80351 
•80334 
.80316 


.73996 
•74041 
.74086 
•74131 
•74176 


1.35142 
1.35060 
1.34978 
1.34896 
1.34814 
1.34732 
1.34650 
1.34568 
1.34487 
1.34405 


.60876 
.60899 
•60922 
•60945 
-60968 


•79335 
•79318 
•79300 
.79282 
•79264 


.76733 
-76779 
-76825 
.76871 
-76918 


1-30323 
1^30244 
1-30166 
1-30087 
1-30009 


30 

29 
28 
27 
26 


35 

36 

37 

38 

39 

40 

41 
!42 

43 
!44 
]45 

46 

47 
U8 
l40 

50 

51 
' 52 

53 

54 

55 

56 

57 
^58 
59_ 
60 


•59599 
.59622 
.59646 
•59669 
.59693, 


.80299 
.80282 
•80264 
•80247 
.80230. 


•74221 

• 74267 
•74312 
•74357 
•74402 

• 74447 
•74492 
•74538 
.74583 
.74628 
.74674 
.74719 
.74764 
.74810 
.74855 


-60991 
-61015 
-61038 
61061 
.61084 


79247 
.79229 
.79211 
.79193 
.79176 


-76964 
•77010 
•77057 
-77103 
•77149 


1-29931 
1-29853 
1-29775 
1^29696 
1- 29518 


25 
24 
23 
22 
21 


.59716 
.59739 
•59763 
•59786 
.59809 


.80212 
•80195 
•80178 
.80160 
•80143 


1-34323 
1-34242 
1-34160 
1-34079 
1.33998 


.61107 
•61130 
•61153 
.31176 
.61199 


.79158 
-79140 
-79122 
-79105 
-79087 


•77196 
•77242 
•77289 
•77335 
.77382 


1-29541 
1-29463 
1-29385 
1.29307 
1^29229 


20 

19 
18 
17 
16 


.59832 
•59856 
•59879 
.59902 
.59926 


.80125 
.80108 
.80091 
.80073 
.80056 


1-33916 
1.33835 
1.33754 
1.33673 
1.33592 


.61222 
•61245 
•61268 
-61291 
-61314 


-79069 
-79051 
.79033 
.79016 
.78998 


-77428 
.77475 
.77521 
.77568 
-77615 


1.29152 
1.29074 
1.28997 
1^28919 
1^ 28842 


15 
14 
13 
12 
11 


.59949 
•59972 
•59995 
•60019 
.60042 


.80038 
.80021 
.80003 
.79986 
.79968 


.74900 
.74946 
.74991 
.75037 
.75082 


1.33511 
1.33430 
1.33349 
1.33268 
1.33187 


•61337 
•61360 
-61383 
-61406 
[-61429 


.78980 
.78962 
.78944 
.78926 
-78908 


-77661 
.77708 
.77754 
.77801 
.77848 


1^ 28764 
1^28687 
1-28610 
1-28533 
1-28456 


10 

9 
8 

7 
6 


•60065 
•60089 
•60112 
•60135 
•60158 


.79951 
.79934 
.79916 
.79899 
.79881 


.75128 
.75173 
.75219 
.75264 
.75310 


1.33107 
1.33026 
1.32946 
1.32865 
1-32785 


-61451 
-61474 
-61497 
•61520 
•61543 


.78891 
-78873 
-78855 
-78837 
78819 


.77895 
.77941 
.77988 
.78035 
.78082 


1-28379 
1-28302 
1-28225 
1-28148 
1-28071 


5 
4 
3 
2 
1 


•60182 


1.79864 


.75355 


1-32704 


•61566 


-78801 


.78129 


1-27994 





/ 


Cos. 


1 Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 








i 


53^ 


7 


13 


1 


i»* 







TABLE IX.--NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS. 
38° SQ"" 



/ 


Sin. 


Cos. 


Tan. 


Cot. 1 


Sin. 


Cos. 


Tan. 


Cot. 


f 




1 

2 
3 
4 


.61566 
.61589 
.61612 
.61635 
.61658 


.78801 
.78783 
.78765 
.78747 
•78729 


.78129 
.78175 
•78222 
•78269 
•78316 




27994 
27917 
27841 
27764 
27688 


•62932 
•62955 
.62977 
•63000 
.63022 


.77715 
.77696 
.77678 
•77660 
.77641 


.80978 
•81027 
.81075 
.81123 
.81171 
.81220 
.81268 
.81316 
.81364 
.81413 


1.23490 
1.23416 
1.23343 
1.23270 
1.23196 


60 

59 
58 
57 
56 


5 
6 
7 
8 
9 


.61681 
•61704 
.61726 
•61749 
.-61772 
•61795 
•61818 
.61841 
.61864 
•61887 


.78711 
.78694 
.78676 
•78658 
•78640 


.78363 
.78410 
.78457 
.78504 
•78551 


1 


27611 
27535 
27458 
27382 
27306 


•63045 
•63068 
•63090 
•63113 
•63135 


.77623 
.77605 
77586 
.77568 
.77550 


1.23123 
1.23050 
1.22977 
1.22904 
1.22831 


55 
54 
53 
52 
51 


10 

11 
12 
13 
14 


•78622 
.78604 
.78586 
•78568 . 
.78550 


•78598 
•78645 
.78692 
.78739 
•78786 


1 


27230 
27153 
27077 
27001 
26925 


•63158 
•63180 
•63203 
•63225 
•63248 


.77531 
.77513 
.77494 
.77476 
.77458 


.81461 
.81510 
.81558 
-81606 
.81655 


1.22758 
1.22685 
1.22612 
1.22539 
1.22467 


50 

49 
48 
47 
46 


15 
16 
17 
18 
19 


•61909 
•61932 
.61955 
•61978 
•62001 


.78532 
.78514 
.78496 
• 78478 
•78460 


•78834 
.78881 
•78928 
•78975 
.79022 




26849 
26774 
26698 
26622 
26546 


•63271 
•63293 
•63316 
■63338 
.63361 


.77439 
.77421 
. 77402 
.77384 
•77366 


.81703 
.81752 
.81800 
.81849 
•81898 


1.22394 
1.22321 
1.22249 
1.22176 
1.22104 


45 
44 
43 
42 
41 


30 

21 
22 
23 
24 


•62024 
•62046 
•62069 
•62092 
•62115 


• 78442 

• 78424 
•78405 
.78387 
.78369 


.79070 
.79117 
•79164 
.79212 
•79259 




26471 
26395 
26319 
26244 
26169 


.63383 
•63406 
•63428 
•63451 
.63473 


•77347 
•77329 
•77310 
•77292 
•77273 


•81946 
.81995 
.82044 
.82092 
.82141 


1.22031 
1.21959 
1.21886 
1.21814 
1.21742 


40 

39 
38 
37 
36 


25 
26 
27 
28 
29 


•62138 
•62160 
•62183 
•62206 
•62229 


.78351 
.78333 
.78315 
.78297 
.78279 


•79306 
.79354 
. 79401 
• 79449 
•79496 




26093 
26018 
25943 
25867 
25792 


•63496 
•63518 
•63540 
•63563 
•63585 


.77255 
.77236 
.77218 
•77199 
•77181 


.82190 
.82238 
.82287 
•82336 
•82385 


1.21670 
1.21598 
1.21526 
1.21454 
1.21382 


35 
34 
33 
32 
31 


30 

31 
32 
33 
34 


•62251 
.62274 
.62297 
.62320 
•62342 


.78261 
.78243 
.78225 
•78206 
•78188 


.79544 
.79591 
•79639 
•79686 
•79734 




25717 
25642 
25567 
25492 
25417 


•63608 
•63630 
.63653 
•63675 
•63698 


.77162 
.77144 
.77125 
.77107 
.77088 


•82434 
•82483 
•82531 
•82580 
•82629 


1.21310 
1.21238 
1-21166 
1.21094 
1.21023 


30 

29 
28 
27 
26 


35 
36 
37 
38 
39 


.62365 
.62388 
.62411 
.62433 
•62456 


.78170 
.78152 
.78134 
.78116 
.78098 


•79781 
•79829 
•79877 
•79924 
•79972 
•80020 
.80067 
.80115 
.80163 
•80211 


-L 


25343 
25268 
25193 
25118 
25044 

24969 
.24895 
.24820 
.24746 
.24672 


-63720 
•63742 
•63765 
•63787 
•63810 

•63832 
•63854 
•63877 
.63899 
.63922 


.77070 
.77051 
.77033 
.77014 
.76996 


•82678 
.82727 
.82776 
.82825 
.82874 


1.20951 
1-20879 
1.20808 
1.20736 
1.20665 


25 
24 
23 
22 
21 


40 

41 
42 
43 
44 


•62479 
•62502 
•62524 
•62547 
•62570 


.78079 
.78061 
.78043 
.78025 
•78007 


•76977 
•76959 
•76940 
•76921 
•76903 


.82923 
.82972 
.83022 
.83071 
.83120 


1.20593 
1.20522 
1.20451 
1.20379 
1.20308 


20 

19 
18 

17 
16 


45 
46 
47 
48 
49 


.62592 
.62615 
.62638 
•62660 
•62683 


.77988 
.77970 
.77952 
.77934 
.77916 


•80258 
•80306 
.80354 
.80402 
•80450 




.24597 
.24523 
24449 
24375 
24301 


•63944 
■63966 
.63989 
.64011 
.64033 


•76884 
•76866 
•76847 
•76828 
•76810 


.83169 
•83218 
•83268 
•83317 
•83366 


1.20237 
1.20166 
1^20095 
1^20024 
1^19953 


15 
14 
13 
12 
11 


50 

51 
52 
53 
54 , 


•62706 
•62728 
•62751 
•62774 
•62796 


•77897 
•77879 
•77861 
.77843 
.77824 


•80498 
•80546 
.80594 
•80642 
•80690 




24227 
24153 
24079 
24005 
23931 


.64056 
.64078 
■64100 
•64123 
.64145 


.76791 
•76772 
.76754 
.76735 
•76717 


•83415 
•83465 
•83514 
•83564 
•83613 


1.19882 
1.19811 
1.19740 
1.19669 
1.19599 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 


•62819 
.62842 
.62864 
•62887 
•62909 


•77806 
•77788 
•77769 
•77751 
•77733 


•80738 
.80786 
•80834 
•80882 
.80930 




23858 
23784 
23710 
23637 
23563 


•64167 
•64190 
.64212 
■64234 
•64256 


•76698 
•76679 
•76661 
•76642 
•76623 


-83662 
•83712 
•83761 
•83811 
•83860 


1.19528 
1.19457 
1.19387 
1.19316 
1.19246 


5 
4 
3 
2 
1 


60 


•62932 


•77715 


80978 


1 


23490 


.64279 


•76604 


•83910 


1.19175 





f 


Cos. 


Sin. 


Cot. 


Tan. 1 


Cos, 


Sin. 


Cot. 


Tan. 


' 



51' 



714 



so'' 



jTABLE IX.- 


-NATURAL SINES, COSINES. TANGENTS, AND COTANGENTS. 
40° 41° 


/ 


Sin. 


Cos. 


Tan. 


Cot. 


Sin. 


Cos. 


Tan. 


Cot. 


/ 


( 

( 1 

^i 

\ 3 
; 4 
( 5 

i 7 
. 8 
i 9 


. 64279 
.64301 
.64323 
.64346 
.64368 


.76604 
.76586 
.76567 
•76548 
•76530 


.83910 
.83960 
•84009 
•84059 
.84108 


1.19175 
1.19105 
1.19035 
1.18964 
1.18894 


65606 
65628 
65650 
.65672 
•65694 


.75471 
.75452 
.75433 
.75414 
.75395 
.75375 
.75356 
.75337 
•75318 
.75299 


86929 
86980 
.87031 
•87082 
•87133 


1.15037 
1.14969 
1.14902 
1.14834 
1.14767 


60 

59 
58 
57 
56 


.64390 
.64412 
.64435 
•64457 
. 64479 


•76511 
.76492 
.76473 
.76455 
.76436 


.84158 
.84208 
.84258 
.84307 
.84357 


1.18824 
1.18754 
1.18684 
1.18614 
1.18544 


•65716 
•65738 
•65759 
.65781 
.65803 


•87184 
•87236 
.87287 
•87338 
•87389 


1.14699 
1.14632 
1.14565 
1.14498 
1 . 14430 


55 
54 
53 
52 
51 


10 

11 

12 

as 

(14 


•64501 
.64524 
. 64546 
.64568 
.64590 


.76417 
.76398 
.76380 
.76361 
.76342 


.84407 
.84457 
.84507 
.84556 
•84606 


1-18474 
1.18404 
1.18334 
1.18264 
1.18194 


.65825 
.65847 
.65869 
.65891 
•65913 


.75280 
.75261 
.75241 
.75222 
.75203 


•87441 
•87492 
•87543 
.87595 
.87646 


1.14363 
1.14296 
1.14229 
1.14162 
1.14095 


50 

49 
48 
47 
46 


(15 
J16 
!17 
$18 


.64612 
.64635 
.64657 
•64679 
.64701 


.76323 
.76304 
.76286 
.76267 
.76248 


.84656 
.84706 
•84756 
•84806 
•84856 


1.18125 
1.18055 
1.17986 
1.17916 
1.17846 


•65935 
•65956 
•65978 
•66000 
•66022 


.75184 
.75165 
.75146 
.75126 
•75107 


.87698 
.87749 
.87801 
.87852 
.87904 


1.14028 
1.13961 
1.13894 
1.13828 
1.13761 


45 
44 
43 
42 
-41 


^30 

(21 
22 
23 
24 


.64723 
.64746 
.64768 
.64790 
•64812 


.76229 
.76210 
.76192 
.76173 
.76154 


•84906 
•54956 
•85006 
•85057 
•85107 


1.17777 
1.17708 
1.17638 
1.17569 
1.17500 


. 66044 
.66066 
.66088 
.66109 
.66131 


.75088 
.75069 
.75050 
.75030 
.75011 


.87955 
.88007 
.88059 
.88110 
•88162 


1.13694 
1.13627 
1.13561 
1.13494 
1.13428 


40 

39 
38 
37 
36 


25 
J26 
27 
28 
29 
30 
31 
32 
33 
34 

135 
136 
37 
38 
39 


.64834 
.64856 
.64878 
.64901 
.64923 


.76135 
.76116 
.76097 
.76078 
•76059 


•85157 
•85207 
•85257 
•85308 
•85358 


1.17430 
1.17361 
1.17292 
1.17223 
1.17154 


.66053 
•66175 
•66197 
.66218 
• 66240 


.74992 
.74973 
.74953 
. 74934 
.74915 


.88204 
.88265 
.88317 
.88369 
.88421 


1-13361 
1^13295 
1^13228 
1.13162 
1.13096 


35 
34 
33 
32 
31 


•64945 
•64967 
•64989 
•65011 
-65033 
•65055 
•65077 
•65100 
.65122 
•65144 


.76041 
.76022 
.76003 
.75984 
.75965 


•85408 
•85458 
•85509 
•85559 
•85609 


1.17085 
1.17016 
1.16947 
1.16878 
1.16809 


.66262 
.66284 
.66306 
.66327 
•66349 


.74896 
.74876 
.74857 
.74838 
•74818 


.88473 
.88524 
.88576 
.88628 
•88680 


1^13029 
1.12963 
1.12897 
1.12831 
1.12765 


30 

29 
28 
27 
26 


.75946 
.75927 
•75908 
•75889 
•75870 


•85660 
•85710 
•85761 
•85811 
•85862 


1.16741 
1.16672 
1.16603 
1.16535 
1.16466 


•66371 
.66393 
. 66414 
.66436 
. 66458 


•74799 
. 74780 
. 74760 

• 74741 

• 74722 


•88732 
•88784 
•88836 
•88888 
•88940 


1.12699 
1.12633 
1.12567 
1.12501 
1.12435 


25 
24 
23 
22 
21 


t40 

41 
42 
43 
44 
' 45 
46 
47 
48 
49 


•65166 
•65188 
•65210 
•65232 
•65254 


•75851 
•75832 
•75813 
•75794 
•75775 


•85912 
•85963 
•86014 
•86064 
.86115 


1.16398 
1.16329 
1.16261 
1.16192 
1.16124 


. 66480 
.66501 
.66523 
.66545 
•66566 


• 74703 

• 74683 

• 74664 

• 74644 
•74G25 


.88992 
.89045 
.89097 
.89149 
.89201 


1.12369 
1.12303 
1.12238 
1.12172 
1.12106 


30 

19 
18 
17 
16 


•65276 
•65298 
• 65320 
•65342 
•65364 


.75756 
.75738 
.75719 
•75700 
•75680 


•86166 
86216 
.86267 
.86318 
•86368 


1.16056 
1.15987 
1.15919 
1.15851 
1.15783 


•66588 
•66610 
•66632 
•66653 
•66675 


• 74606 
•74586 
•74567 

• 74548 

• 74528 


.89253 
.89306 
.89358 
.89410 
.89463 


1.12041 
1.11975 
1.11909 
1.11844 
1.11778 


15 
14 
13 
12 
11 


50 

51 

1 52 

J 53 

54 


•65386 
•65408 
•65430 
•65452 
•65474 


•75661 
.75642 
.75623 
.75604 
.75585 


•86419 
•86470 
•86521 
•86572 
•86623 


1.15715 
1.15647 
1.15579 
1.15511 
1.15443 


•66697 
•66718 
•66740 
•66762 
•66783 


. 74509 
. 74489 
. 74470 
.74451 
•74431 


.89515 
.89567 
.89620 
.89672 
.89725 


1^11713 
1.11648 
1.11582 
1.11517 
1.11452 


10 

9 
8 
7 
6 


55 
56 
57 
58 
59 
60 


•65496 
.65518 
.65540 
.65562 
.65584 


.75566 
.75547 
.75528 
.75509 
.75490 


•86674 
•86725 
•86776 
•86827 
•86878 


1.15375 
1.15308 
1.15240 
1.15172 
1.15104 


•66805 
•66827 
•66848 
•66870 
•66891 


.74412 
.74392 
.74373 
.74353 
.74334 
.74314 


.89777 
.89830 
.89883 
.89935 
.89988 


1.11387 
1.11321 
1.11256 
1.11191 
1.11126 


5 

4 
3 

2 

1 


.65606 


.75471 


•86929 


1.15037 


.66913 


.90040 


1.11061 





/ 


Cos. 


Sin. 


Cot. 


Tan. 


Cos. 


Sin. 


Cot. 


Tan. 


/ 


1 




^ 


19« 


71 


5 




48^ 







TABLE IX.- 


-NATURAL SINES, COSINES, TANGENTS, AND COTANGENIH 
43° 43° 


9 


Sin. 


Cos. 


Tan. 


Cot, 


Sin. 


Cos. 


Tan. 


Cot. 1 


f 




1 

2 
3 

4 


.66913 
-66935 
.66956 
.66978 
.66999 


.74314 
.74295 
.74276 
.74256 
•74237 


.90040 
.90093 
.90146 
.90199 
.90251 


1.11061 
1 10996 
1.10931 
1.10867 
1.10802 


•68200 
.68221 
.68242 
.68264 
.68285 


•73135 
.73116 
.73096 
.73076 
.73056 


.93252 
.93306 
.93360 
.93415 
.93469 




07237 
07174 
07112 
07049 
06987 


60 

59 
58 
57 
56 
55 
54 
53 
52 
51 


5 
6 
7 
8 
9 


.67021 
67043 
.67064 
.67086 
.67107 


. 74217 
.74198 
74178 
•74159 
.74139 


.90304 
.90357 
.90410 
.90463 
.90516 


1.10737 
1.10672 
1.10607 
1.10543 
1.10478 


68306 
.68327 
.68349 
.68370 
.68391 


.73036 
.73016 
.72996 
.72976 
.72957 


.93524 
.93578 
.93633 
.93688 
.93742 




06925 
06862 
06800 
06738 
06676 


10 

11 

12 
13 
14 


.67129 
.67151 
.67172 
.67194 
.67215 


.74120 
.74100 
. 74080 
.74061 
. 74041 


.90569 
.90621 
.90674 
90727 
.90781 


1.10414 
1.10349 
1.10285 
1.10220 
1.10156 


.68412 
. 68434 
.68455 
. 68476 
. 68497 


.72937 
.72917 
.72897 
.72877 
.72857 


.93797 
.93852 
.93906 
.93961 
.94016 




06613 
06551 
06489 
06427 
06365 


50 

49 
48 
47 
46 1 


15 
16 
17 
18 
19 


.67237 
.67258 
.67280 
.67301 
.67??23 
. 67344 
.67366 
.67387 
. 67409 
.67430 


. 74022 
. 74002 
.73983 
.73963 
.73944 

. 73924 
. 73904 
. 73885 
.73865 
.73846 


.90834 
.90887 
.90940 
.90993 
.91046 

.91099 
.91153 
.91206 
.91259 
.91313 


1.10091 
1.10027 
1.09963 
1.09899 
1.09834 


.68518 
.68539 
.68561 
.68582 
. 68603 


.72837 
.72817 
.72797 
.72777 
.72757 


.94071 
.94125 
.94180 
.94235 
.94290 




06303 
06241 
06179 
06117 
06056 


45 
44 
43 
42 
_ 41 1 


20 

21 
22 
23 
24 


1^09770 
1.09706 
1.09642 
1.09578 
1.09514 


.68624 
.68645 
.68666 
.68688 
.68709 


.72737 
.72717 
•72697 
.72677 
.72657 


.94345 
. 94400 
. 94455 
.94510 
.94565 




05994 
05932 
05870 
05809 
05747 


40 

39! 
38 
37 
36 


25 
26 
27 
28 
29 


67452 
.67473 
.67495 
.67516 
.67538 


.73826 
.73806 
.73787 
.73767 
•73747 


.91366 
.91419 
.91473 
.91526 
•91580 


1.09450 
1.09386 
1.09322 
1.09258 
1.09195 


.68730 
.68751 
.68772 
•68793 
.68814 


.72637 
.72617 
.72597 
.72577 
.72557 


.94620 
.94676 
.94731 
.94786 
.94841 




05685 
05624 
05562 
05501 
05439 


35 

33 
32 
31 


30 

31 
32 
33 
34 


.67559 
.67580 
.67602 
•67623 
.67645 


.73728 
.73708 
•73688 
•73669 
.73649 


.91633 
.91687 
.91740 
•91794 
•91847 


1.09131 
1.09067 
1.09003 
1.08940 
1.08876 


.68835 
•68857 
.68378 
.68899 
.68920 


.72537 
.72517 
.72497 
. 72477 
. 72457 

.72437 
.72417 
.72397 
.72377 
.72357 


.94896 
.94952 
•95007 
•95062 
-95118 


1 


05378 
05317 
05255 
05194 
05133 


30 

29 
28 
27 
26 1 


35 
36 
37 
38 


.67666 
.67688 
.67709 
.67730 
67752 


.73629 
.73610 
•73590 
73570 
•73551 


•91901 
.91955 
.92008 
•92062 
•92116 


1.08813 
1.08749 
1.08686 
1.08622 
!• 08559 


. 68941 
•68962 

• 68983 

• 69004 
•69025 


•95173 
•95229 
.95284 
.95340 
.95395 




05072 
05010 
04949 
04888 
04827 


25 

24 
23 
22 
21 i 


40 

41 
42 
43 
44 


.67773 
.67795 
.67816 
.67837 
.67859 


.73531 
•73511 
.73491 
.73472 
.73452 


•92170 
.92224 
.92277 
•92331 
•92385 


1.08496 
1.08432 
1.08369 
1.08306 
1.08243 


.69046 
•69067 
•69088 
•69109 
.69130 


.72337 
.72317 
.72297 
.72277 
.72257 
.72236 
•72216 
.72196 
.72176 
.72156 


.95451 
.95506 
.95562 
.95618 
.95673 




04766 
04705 
04644 
04583 
04522 


30 j 

19 

18 

17 

16 

15 1 

14 J 

13 

121 

ll.i 


45 
46 
47 
48 
49 


.67880 
.67901 
.67923 
. 67944 
.67965 


. 73432 
.73413 
•73393 
•73373 
•73353 


•92439 
.92493 
.92547 
.92601 
•92655 


1.08179 
1.08116 
1.08053 
1.07990 
1.07927 


•69151 
.69172 
.69193 
•69214 
•69235 


.95729 
.95785 
.95841 
.95897 
•95952 
•96008 
•96064 
•96120 
•96176 
•96232 




04461 
04401 
04340 
04279 
.04218 


50 

51 
52 
53 
54 


.67987 

. 68008 

.68029 

68051 

68072 

68093 
.68115 
.68136 
•68157 
.68179 
,68200 
Cos. 


•73333 
•73314 
•73294 
.73274 
• 73254 


•92709 
•92763 
.92817 
.92872 
.92926 
.92980 
•93034 
.93088 
•93143 
•93197 
•93252 
Cot. 


!• 07864 
1.07801 
1. 07738 
1.07676 
1.07613 


.69256 
•69277 
•69298 
•69319 
•69340 


•72136 
.72116 
•72095 
.72075 
.72055 




.04158 
.04097 
.04036 
.03976 
.03915 
.03855 
.03794 
•03734 
•03674 
.03613 


10'! 

si 

^; 

6 
51 
4 
3 
21 


55 
56 
57 
58 
59 


•73234 
.73215 
.73195 
.73175 
.73155 


1.07550 
1.07487 
1.07425 
1.07362 
1.07299 


.69361 
-69382 
• 69403 
. 69424 
-69445 


.72035 
.72015 
•71995 
.71974 
-71954 


•96288 
-96344 
•96400 
96457 
•96513 


60 


.73135, 
Sin. 


1-07237 
Tan. 


69466 
Cos. 


71934 
Sin. 


.96569 
Cot. 


J^ 


-03553 
Tan. 


-t2 






< 


17* 


7 


16 


4 


16** 









TABLE IX.~NATURAL SINES, COSINES. TANGENTS. AND COTANGENTS 
44° 44° 



^ / 


Sin. 


Cos. 


Tan. 


Cot. 


f 


t 


Sin. 


Cos. 


Tan- 


Cot. 


/ 


B 
\ 1 
£ 2 

c 3 
£4 

: 5 

I 6 
f 7 
3 8 

I 9 


•69466 
•69487 
.69508 
•69529 
•69549 


•71934 
•71914 
.71894 
•71873 
•71853 


.96569 
•96625 
.96681 
.96738 
.96794 


1-03553 
1- 03493 
1^ 03433 
1^03372 
1^03312 


60 

59 
58 
57 
56 


30 

31 

32 
33 
34 
35 
36 
37 
38 
39 


.70091 
.70112 
-70132 
.70153 
.70174 


.71325 
.71305 
.71284 
.71264 
•71243 


•98270 
.98327 
.98384 
.98441 
•98499 


1.01761 
1.01702 
1^01642 
1.01583 
1.01524 


30 

29 
28 
27 
26 


•69570 
•69591 
•69612 
•69633 
•69654 


.71833 
.71813 
.71792 
.71772 
.71752 


.96850 
•96907 
.96963 
•97020 
.97076 


1.03252 
1.03192 
1.03132 
1.03072 
1.03012 


55 
54 
53 
52 
51 


.70195 
.70215 
.70236 
.70257 
•70277 


•71223 
•71203 
.71182 
.71162 
.71141 


.98556 
.98613 
.98671 
•98728 
•98786 


1.01465 
1.01406 
1.01347 
1.01288 
1.01229 


25 
24 
23 
22 
21 


M2 
^3 
J14 


.69675 
•69696 
.69717 
.69737 
.69758 


.71732 
.71711 
.71691 
.71671 
.71650 


•97133 
.97189 
.97246 
.97302 
.97359 


1 02952 
1^02892 
1.02832 
1.02772 
1.02713 

1.02653 
1.02593 
1.02533 

1.02474 
1.02414 


50 

49 
48 
47 
46 


40 

41 
42 
43 
44 


.70298 
•70319 
•70339 
•70360 
.70381 


.71121 
.71100 
.71080 
.71059 
.71039 


98843 
.98901 
.98958 
.99016 
.99073 


1.01170 
1.01112 
1.01053 
1.00994 
1.00935 


20 

19 
18 
17 
16 


as 

{16 


•69779 
.69800 
.69821 
.69842 
.69862 


.71630 
.71610 
.71590 
.71569 
•71549 


.97416 
.97472 
.97529 
.97586 
.97643 


45 
44 
43 
42 

,^1 


45 
46 
47 
48 
49 


. 70401 
. 70422 
70443 
. 70463 
. 70484 


.71019 
.70998 
.70978 
.70957 
70937 
.70916 
.70896 
.70875 
•70855 

• 70834 

• 70813 
•70793 
.70772 
. 70752 
.70731 


•99131 
.99189 
.99247 
.99304 
•99362 


1.00876 
1.00818 
1.00759 
1.00701 
1.00642 


15 
14 
13 
12 
11 


if 

E@2 
!(23 

1 

129 


.69883 
.69904 
.69925 
•69946 
.69966 


.71529 
.71508 
.71488 
.71468 
.71447 


•97700 
.97756 
.97813 
.97870 
.97927 


1-02355 
1.02295 
1.02236 
1.02176 
1-02117 


40 

39 
38 
37 
36 


50 

51 
52 
53 
54 


.70505 
.70525 
.70546 
.70567 
.70587 


.99420 
.99478 
.99536 
.99594 
.99652 


1.00583 
1.00525 
1.00467 
1^00408 
1^00350 


lO 
9 
8 
7 

6 


•69987 
• 70008 
•70029 
.70049 
.70070 


.71427 
.71407 
.71386 
.71366 
.71345 


.97984 
98041 
.98098 
.98155 
.98213 


1.02057 
1.01998 
1.01939 
1.01879 
1.01820 


35 
34 
33 
32 
31 
30 


55 
56 
57 
58 
59 


. 70608 
.70628 
. 70649 
.70670 
.70690 


.99710 
.99768 
.99826 
.99884 
.99942 


1.00291 
1.00233 
1.00175 
1.00116 
1.00058 


5 

4 
3 

2 

1 


.70091 

Cos. 


•71325 


-98270 


1.01761 


60 


.70711 


•70711 


1.00000 


1.00000 


_J^ 


Sin. 


Cot. 


Tan. 


' 


' 


Cos. 


Sin, 


Cot. 


Tan. ' 



45° 



717 



45^ 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS, 
0° 1° 3° 3° 


t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


f 




1 
2 
3 
4 


.00000 
.00000 
.00000 
.00000 
.00000 


.00000 
.00000 
.00000 
.00000 
.00000 


.00015 
.00016 
.00016 
.00017 
.00017 


.00015 
.00016 
.00016 
.00017 
.00017 


.00061 
.00062 
.00063 
.00064 
.00065 


.00061 
.00062 
.00063 
.00064 
.00065 


•00137 
•00139 
.00140 
.00142 
.00143 


.00137 
.00139 
.00140 
.00142 
.00143 




1 
2 
3 

4 \ 


5 
6 
7 
8 
9 


.00000 
.00000 
.00000 
.00000 
.00000 


.00000 
.00000 
.00000 
.00000 
.00000 


.00018 
.00018 
.00019 
.00020 
^00020 


.00018 
.00018 
.00019 
.00020 
.00020 


.00066 
.00067 
.00068 
.00069 
.00070 


.00066 
.00067 
.00068 
.00069 
.00070 


.00145 
.00146 
.00148 
•00150 
•00151 


.00145 
.00147 
.00148 
.00150 
.00151 


5l 

6 

7 

8 

9 


10 

11 
12 
13 
14 


.00000 
.00001 
.00001 
.00001 
.00001 


.00000 
.00001 
.00001 
.00001 
.00001 


.00021 
.00021 
.00022 
.00023 
.00023 


.00021 
.00021 
.00022 
.00023 
.00023 


.00071 
.00073 
.00074 
.00075 
•00076 


.00072 
.00073 
.00074 
.00075 
.00076 


.00153 
.00154 
•00156 
.00158 
.00159 


.00153 
.00155 
.00156 
.00158 
.00159 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.00001 
.00001 
.00001 
.00001 
.00002 


.00001 
.00001 
.00001 
.00001 
.00002 


.00024 
.00024 
•00025 
.00026 
.00026 


.00024 
.00024 
.00025 
.00026 
.00026 


•00077 
•00078 
•00079 
•00081 
•00082 


.00077 
.00078 
.00079 
.00081 
•00082 


.00161 
.00162 
.00164 
.00166 
.00168 


.00161 
.00163 
.00164 
.00166 
.00168 


15 
16 
17 
18 
19 
30 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 


30 

21 
22 
23 
24 


.00002 
. 00002 
00002 
.00002 
•00002 


.00002 
.00002 
.00002 
.00002 
-00002 


.00027 
.00028 
.00028 
.00029 
.00030 


.00027 
.00028 
.00028 
.00029 
.00030 


•00083 
•00084 
00085 
•00087 
•00088 


.00083 
.00084 
.00085 
.00087 
.00088 


.00169 
.00171 
.00173 
.00174 
.00176 


.00169 
.00171 
.00173 
.00175 
•00176 


25 
26 
27 
28 
29 


.00003 
00003 
•00003 
.00003 
.00004 


.00003 
.00003 
.00003 
.00003 
.00004 


.00031 
•00031 
.00032 
.00033 
00034 


.00031 
.00031 
.00032 
.00033 
.00034 


•00089 
-00090 
•00091 
•00093 
•00094 


.00089 
.00090 
.00091 
.00093 
.00094 


.0017d 
•00179 
•00181 
•00183 
•00185 


.00178 
.00180 
.00182 
•00183 
•00185 


30 

31 
32 
33 
34 


.00004 
.00004 
.00004 
.00005 
.00005 


.00004 
.00004 
.00004 
.00005 
.00005 


•00034 
.00035 
.00036 
.00037 
.00037 


.00034 
.00035 
•00036 
.00037 
.00037 


00095 
•00096 
•00098 
•00099 
•00100 


.00095 
•00097 
•00098 
.00099 
.00100 


00187 

00188 

•00190 

•00192 

•00194 


.00187 
.00189 
.00190 
•00192 
•00194 


35 
36 
37 
38 
39 


00005 
.00005 
.00006 
.00006 
.00006 


.00005 
.00005 
.00006 
00006 
.00006 


.00038 
.00039 
.00040 
.00041 
.00041 


.00038 
.00039 
•00040 
.00041 
.00041 


•00102 
•00103 
•00104 
•00106 
•00107 


.00102 
.00103 
.00104 
.00106 
.00107 


•00196 
•00197 
•00199 
•00201 
•00203 


.00196 
.00198 
.00200 
.00201 
•00203 


40 

41 
42 
43 
44 


.00007 
00007 
•00007 
•00008 
.00008 


.00007 
.00007 
.00007 
.00008 
•00008 


.00042 
.00043 
•00044 
.00045 
.00046 


•00042 
.00043 
.00044 
.00045 
.00046 


•00108 
•00110 
•00111 
•00112 
•00114 


.00108 
.00110 
•00111 
•00113 
•00114 


•00205 
•00207 
•00208 
•00210 
■00212 


.00205 
•00207 
.00209 
.00211 
•00213 


40 

41 
42 
43 
44 


45 
46 
47 
47 
49 


.00009 
00009 
.00009 
.00010 
00010 


-00009 
.00009 
.00009 
.00010 
.00010 


.00047 
00048 
•00048 
•00049 
•00050 


.00047 
.00048 
.00048 
.00049 
.00050 


•00115 
• 00U7 
•00118 
•00119 
•00121 


.00115 
•00117 
.00118 
.00120 
.00121 


•00214 
•00216 
•00218 
•00220 
• 00222 


00215 
.^0216 
.00218 
.00220 
.00222 


45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 


60 

51 
52 
53 
54 


•00011 
00011 
•00011 
•00012 
•00012 


.00011 
.00011 
.00011 
.00012 
.00012 


•00051 
.00052 
•00053 
•00054 
•00055 


.00051 
.00052 
.00053 
.00054 
.00055 


•00122 
•00124 
•00125 
•00127 
•00128 


.00122 
.00124 
.00125 
.00127 
•00128 


•00224 
•00226 
•00228 
00230 
■00232 


.00224 
.00226 
.00228 
.00230 
•00232 
.00234 
.00236 
•00238 
.00240 
•00242 


55 
56 
57 
58 
59 


•00013 
•00013 
•00014 
•00014 
00015 


.00013 
.00013 
.00014 
.00014 
.00015 


00056 
•00057 
.00058 
•00059 

00060 


.00056 
.00057 
•00058 
•00059 
00060 


00130 
•00131 
•00133 
•00134 
•00136 


.00130 
.00131 
00133 
.00134 
•00136 


•00234 
•00236 
00238 
•00240 
•00242 


60 


•00015 


.00015 


•00061 


.00061 


00137 


•00137 


.00244 .00244 60 



718 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS 
4° 5** 6° 7° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
8 
4 


.00244 
•00246 
•00248 
.00250 
•00252 


.00244 
.00246 
.00248 
.00250 
.00252 


.00381 
00383 
.00386 
.00388 
.00391 


.00382 
.00385 
.00387 
.00390 
.00392 


.00548 
.00551 
.00554 
.00557 
.00560 


.00551 
.00554 
.00557 
.00560 
•00563 


00745 
.00749 
-00752 
.00756 
•00760 


.00751 
.00755 
.00758 
.00762 
•00765 




1 
2 
3 
4 


5 

6 

I 7 

I 8 

? 9 


•00254 
.00256 
.00258 
.00260 
.00262 


.00254 
.00257 
.00259 
.00261 
00263 


.00393 
.00396 
.00398 
.00401 
.00404 


.00395 
.00397 
.00400 
.00403 
.00405 


.00563 
.00566 
.00569 
.00572 
.00576 


.00566 
.00569 
•00573 
.00576 
.00579 


■00763 
•00767 
•00770 
.00774 
.00778 


•00769 
•00773 
•00776 
•00780 
•00784 


5 
6 
7 
8 
9 


10 

I 11 
S 12 
' 13 

ti 14 


00264 
.00266 
.00269 
• 00271 

00273 


.00265 
.00267 
.00269 
.00271 
.00274 


.00406 
.00409 
•00412 
.00114 
.00417 
.00420 
.00422 
.00425 
.00428 
•00430 


.00408 
.00411 
.00413 
.00416 
.00419 


00579 
.00582 
00585 
00588 
.00591 


.00582 
.00585 
.00588 
.00592 
.00595 


.00781 
.00785 
.00789 
00792 
.00796 


•00787 
.00791 
•00795 
•00799 
.00802 


10 

11 
12 
13 
14 


c 15 
e 16 
^ 17 
i 18 

ei9 


•00275 
.00277 
•00279 
•00281 
• 00284 


.00276 
.00278 
.00280 
.00282 
.00284 


.00421 
.00424 
.00427 
.00429 
.00432 


00594 
.00598 
•00601 
.00604 
•00607 


.00598 
.00601 
.00604 
.00608 
•00611 


.00800 
. 00803 
.00807 
.00811 
.00814 


.00806 
.00810 
.00813 
•00817 
.00821 


15 
16 
17 
18 
19 


(20 
121 
^ 22 
2 23 

^24 


• 0028.6 
.0028S 
•00290 
•00293 
.00295 


.00287 
.00289 
.00291 
.00293 
.00296 


•00433 
•00436 
.00438 
•00441 
•00444 


.00435 
.00438 
.00440 
.00443 
.00446 


.00610 
-00614 
.00617 
.00620 
•00623 


00614 
.00617 
.00621 
.00624 
.00627 


.00818 
.00822 
.00825 
•00829 
.00833 


•00825 
•00828 
•00832 
.00836 
•00840 


20 

21 
22 
23 
24 


a 25 
:i'26 
/27 
^28 
C29 


•00297 
•00299 
.00301 
•00304 
•00306 


.00298 
.00300 
.00302 
.00305 
.00307 


.00447 
.00449 
.00452 
.00455 
.00458 


.00449 
.00451 
.00454 
.00457 
.00460 


•00626 
•00630 
•00633 
•00636 
• 00640 


.00630 
.00634 
.00637 
.00640 
.00644 


•00837 
.00840 
.00844 
.00848 
.00852 


.00844 
•00848 
.00851 
.00855 
.00859 


25 
26 
27 
28 
29 


130 

rsi 

l!32 
i33 

334 


•00308 
•00311 
•C0313 
00315 
•00317 


.00309 
.00312 
.00314 
.00316 
.00318 


.00460 
.00463 
.00466 
.00469 
.00472 


.00463 
.00465 
.00468 
.00471 
.00474^ 


• 00643 
•00646 
•00649 
•00653 
.00656 


.00647 
.00650 
.00654 
.00657 
.00660 


.00856 
00859 
.00863 
.00867 
.00871 


.00863 
.00867 
.00871 
.00875 
.00878 


30 

31 
32 
33 
34 


J35 
\U6 
137 
538 

;!39 


•00320 
•00322 
.00324 
•00327 
•00329 


.00321 
.00323 
.00328 
.00328 
.00330 


.00474 
.00477 
.00480 
•00483 
•00486 


.00477 
.00480 
.00482 
.00485 
.00488 


•00659 
.00663 
.00666 
.00669 
•00673 


.00664 
.00667 
.00671 
.00674 
.00677 


.00875 
.00878 
•00882 
•00886 
•00890 


.00882 
.00886 
.00890 
.00894 
.00898 


35 
36 
37 
38 
39 


ko 

41 

:!!42 

343 
k4 
45 
46 
47 
48 
]49 


•00332 
•00334 
•00336 
•00339 
.00341 


.00333 
.00335 
.00337 
.00340 
.00342 


.00489 
.00492 
.00494 
.00497 
•00500 


•00491 
.00494 
.00497 
.00500 
.00503 


•00676 
.00680 
•00683 
•00686 
•00690 


.00681 
•00684 
•00688 
.00691 
•00695 


•00894 
•00898 
.00902 
.00906 
.00909 


.00902 
.00906 
.00910 
.00914 
•00918 


40 

41 
42 
43 
44 


.00343 
.00346 
.00348 
00351 
00353 


.00345 
.00347 
00350 
.00352 
.00354 


•00503 
.00506 
.00509 
.00512 
.00515 


.00506 
.00509 
.00512 
•00515 
.00518 


.00693 
00607 
•00700 
•00703 
•00707 


.00698 
.00701 
.00705 
•00708 
•00712 


.00913 
00917 
00921 
00925 

•00929 


.00922 
.00926 
.00930 
.00934 
•00938 


45 
46 
47 
48 
49 


^0 

;5i 

152 
53 
?54 


•00356 
•00358 
00361 
.00363 
.00365 


.00357 
.00359 
.00362 
.00364 
.00367 


.00518 
.00521 
.00524 
.00527 
.00530 


•00521 
.00524 
.00527 
.00530 
.00533 


.00710 
.00714 
.00717 
.00721 
.00724 


.00715 
.00719 
•00722 
.00726 
.00730 


.00933 
00937 
.00941 
.00945 
•00949 


.00942 
.00946 
.00950 
.00954 
.00958 


50 

51 
52 
53 
54 


55 
■56 
|57 

ji 


•00368 
.00370 
•00373 
•00375 
•00378 


.00369 
.00372 
•00374 
■00377 
.00379 


.00533 
.00536 
.00539 
.00542 
00545 


.00536 
.00539 
.00542 
.00545 
00548 


.00728 
.00731 
.00735 
•00738 
.00742 


.00733 
.00737 
.00740 
•00744 
.00747 


•00953 
•00957 
•00961 
•00965 
•00969 


.00962 
.00966 
.00970 
.00975 
.00979 


55 
56 
57 
58 
59 


'30 


•00381 


.00382 


00548 


.00551 


.00745 


.00751 


.00973 


.00983 


60 


1 








7 


19 











TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
8° 9° 10° 11*» 


/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 

4 


.00973 
.00977 
.00981 
.00985 
.00989 


.00983 
.00987 
.00991 
.00995 
.00999 


.01231 
.01236 
.01240 
.01245 
.01249 




01247 
01251 
01256 
01261 
01265 


•01519 
.01524 
.01529 
.01534 
.01540 


•01543 
.01548 
.01553 
.01558 
•01564 


•01837 
.01843 
.01848 
•01854 
•01860 


.01872 
.01877 
.01883 
.01889 
•01895 



1 

2 
3 

4 


5 
6 
7 
8 
9 


00994 
•00998 
.01002 
.01006 
.01010 


.01004 
.01008 
.01012 
.01016 
.01020 


.01254 
.01259 
.01263 
•01268 
•01272 




01270 
01275 
01279 
01284 
01289 


.01545 
.01550 
.01555 
.01560 
.01565 


.01569 
.01574 
.01579 
.01585 
•01590 


•01865 
01871 
•01876 
•01882 
•01888 


.01901 
.01906 
.01912 
.01918 
•01924 


5 

6 

7 

8 

9 

10 

11 

12 

13 

14 


10 

11 
12 
13 

14 


.01014 
.01018 
.01022 
.01027 
.01031 


.01024 
.01029 
.01033 
.01037 
.01041 


.01277 
.01282 
.01286 
.01291 
•01296 




01294 
01298 
01303 
01308 
01313 


.01570 
.01575 
.01580 
.01586 
.01591 


.01595 
.01601 
.01606 
.01611 
.01616 


•01893 
•01899 
•01904 
•01910 
•01916 


•01930 
•01936 
•01941 
•01947 
•01953 


15 
16 
17 
18 
19 


.01035 
•01039 
•01043 
•01047 
•01052 


.01046 
.01050 
.01054 
.01059 
.01063 


.01300 
.01305 
•01310 
•01314 
01319 




01318 
01322 
01327 
01332 
01337 


•01596 
•01601 
.01606 
•01612 
•01617 


.01622 
.01627 
.01633 
.01638 
.01643 


•01921 
•01927 
•01933 
•01939 
.01944 


•01959 
.01965 
.01971 
.01977 
•01983 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


•01056 
01060 
01064 
•01069 
•01073 


.01067 
.01071 
.01076 
.01080 
.01084 


•01324 
.01329 
.01333 
•01338 
.01343 




01342 
01346 
01351 
01356 
01361 


•01622 
•01627 
•01632 
•01638 
•01643 


•01649 
•01654 
.01659 
.01665 
.01670 


•01950 
•01956 
•01961 
•01967 
.01973 


•01989 
.01995 
.02001 
•02007 
.02013 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.01077 
.01081 
.01086 
•01090 
•01094 


.01089 
.01093 
.01097 
.01102 
.01106 


.01348 
.01352 
.01357 
.01362 
.01367 
.01371 
.01376 
.01381 
•01386 
.01391 




01366 
01371 
01376 
01381 
01386 


•01648 
•01653 
•01659 
•01664 
.01669 


.01676 
.01681 
.01687 
.01692 
.01698 


•01979 
.01984 
•01990 
.01996 
.02002 


•02019 
•02025 
•02031 
•02037 
.02043 


25 
26 
27 
28 
29 
30 
31 
32 
33 
34 


30 

31 
32 
33 
34 


•01098 
.01103 
•01107 
.01111 
•01116 


.01111 
.01115 
.01119 
.01124 
.01128 




01391 
01395 
01400 
01405 
01410 


•01675 
.01680 
.01685 
.01690 
•01696 


.01703 
.01709 
.01714 
.01720 
.01725 


.02008 
•02013 
•02019 
•02025 
•02031 


•02049 
•02055 
•02061 
•02067 
•02073 


35 
36 
37 
38 
39 


•01120 
•01124 
•01129 
•01133 
•01137 

01142 
.01146 
.01151 
•01155 
•01159 


.01133 
.01137 
.01142 
.01146 
•01151 

.01155 
.01160 
.01164 
.01169 
01173 


•01396 
.01400 
.01405 
.01410 
.01415 

■01420 
.01425 
•01430 
-01435 
.01439 




01415 
01420 
01425 
01430 
01435 


.01701 
.01706 
.01712 
•01717 
•01723 
•01728 
01733 
.01739 
•01744 
.01750 


•01731 

.01736 

.01742 

.01747 • 

.01753 

.01758 

.01764 

.01769 

.01775 

.01781 


•02037 
•02042 
•02048 
.02054 
-02060 
.02066 
.02072 
.02078 
•02084 
•02090 


•02079 
•02085 
•02091 
•02097 
.02103 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 




01440 
01445 
01450 
01455 
01461 


•02110 
•02116 
•02122 
•02128 
•02134 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


.01164 
• 01168 
01173 
•01177 
•01182 


.01178 
.01182 
.01187 
.01191 
.01196 


.01444 
.01449 
.01454 
.01459 
•01464 




.01466 
01471 
.01476 
.01481 
01486 


.01755 
.01760 
01766 
•01771 
•01777 


01786 
.01792 
.01793 
.01803 
•01809 


•02095 
•02101 
•02107 
•02113 
•02119 


•02140 
•02146 
•02153 
•02159 
•02165 


45 
46 
47 
43 
49 


50 

51 
R2 
53 
54 


•01186 
•01191 
•01195 
•01200 
■01204 


.01200 
.01205 
.01209 
.01214 
.01219 


•01469 
.01474 
.01479 
.01484 
.01489 




01491 
01496 
01501 
01506 
01512 


.01782 
.01788 
.01793 
•0179b 
•01804 


•01815 
.01820 
.01826 
.01832 
01837 


•02125 
•02131 
•02137 
•02143 
•02149 


.02171 
.02178 
.02184 
•02190 
•02196 


50 

51 
52 
53 
54 


55 
56 
57 
58 
.59. 


•01209 
•01213 
•01218 
-01222 
•01227 


.01223 
.01228 
.01233 
.01237 
.01242 


.01494 
.01499 
.01504 
.01509 
01514 




01517 
01522 
01527 
01532 
01537 


•01810 
•01815 
•01821 
•01826 
.01832 


•01843 
•01849 
.01854 
.01860 
.01866 


•02155 
.02161 
•02167 
•02173 
•02179 


•02203 
•02209 
•02215 
•02221 
.02228 


55 
56 
57 
58 
59 


60 


01231 


.01247 


.01519 




01543 


.01837 


.01872 


•02185 


.02234 


60 



720 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
13° 13° 14° 15° 



» 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. see. 


/ 




1 

2 
3 
4 


.02185 
.02191 
•02197 
.02203 
.02210 


.02234 
.02240 
.02247 
.02253 
.02259 


.02563 
•02570 
•02576 
•02583 
.02589 


.02630 
.02637 
.02644 
.02651 
.02658 


02970 
.02977 
.02985 
•02992 
•02999 


.03061 
.03069 
.03076 
•03084 
.03091 


•03407 
•03415 
.03422 
.03430 
•03438 


.03528 
.03536 
.03544 
.03552 
•03560 
.03568 
.03576 
.03584 
.03592 
.03601 




3 

4 


5 
6 
7 
8 
9 


.02216 
.02222 
.02228 
.02234 
.02240 
.02246 
.02252 
.02258 
.02265 
.02271 


.02266 
.02272 
.02279 
.02285 
.02291 


•02596 
.02602 
.02609 
.02616 
.02622 


.02665 
.02672 
.02679 
.02686 
.02693 


•03006 
•03013 
•03020 
•03027 
.03034 


.03099 
.03106 
.03114 
.03121 
.03129 


.03445 
•03453 
03460 
•03468 
•03476 


5 
6 
7 
8 

9 


10 

11 
12 
13 
14 


.02298 
•02304 
.02311 
.02317 
.02323 


.02629 
•02635 
•02642 
.02649 
.02655 


.02700 
.02707 
.02714 
.02721 
.02728 


.03041 
.03048 
.03055 
.03063 
•03070 


.03137 
.03144 
.03152 
.03159 
.03167 


•03483 
•03491 
•03498 
•03506 
•03514 


.03609 
.03617 
.03625 
.03633 
•03642 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


02277 
.02283 
-02289 
.02295 
•02302 


.02330 
.02336 
.02343 
.02349 
.02356 


.02662 
.02669 
.02675 
.02682 
.02689 


.02735 
.02742 
.02749 
.02756 
.02763 


.03077 
.03084 
.03091 
.03098 
•03106 


.03175 
.03182 
.03190 
•03198 
•03205 


•03521 
•03529 
.03537 
.03544 
•03552 


.03650 
.03658 
.03666 
.03674 
.03683 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


.02308 
.02314 
•02320 
.02327 
•02333 


.02362 
.02369 
.02375 
•02382 
.02388 


.02696 
.02702 
•02709 
.02716 
•02722 


.02770 
.02777 
.02784 
.02791 
.02799 


.03113 
.03120 
.03127 
.031S4 
.03142 


•03213 
•03221 
•03228 
.03236 
.03244 
.03251 
.03259 
.03267 
•03275 
•03282 


.03560 
.03567 
.03575 
•03583 
.03590 


.03691 
.03699 
.03708 
•03716 
•03724 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•02339 
.02345 
•02352 
•02358 
.02364 


•02395 
•02402 
.02408 
.02415 
.02421 


•02729 
.02736 
.02743 
•02749 
•02756 


.02806 
.02813 
.02820 
.02827 
.02834 


•03149 
.03156 
•03163 
•03171 
•03178 


.03598 
•03606 
.03614 
•03621 
.03629 


•03732 
.03741 
.03749 
.03758 
•03766 


2§ 
26 
27 
23 
29 


30 

31 
32 
33 
34 


•02370 
•02377 
.02383 
.02389 
02396 


•02428 
.02435 
.02441 
.02448 
•02454 


•02763 
•02770 
•02777 
•02783 
.02790 


.02842 
.02849 
.02856 
.02863 
.02870 


-03185 
•03193 
.03200 
•03207 
•03214 


•03290 
•03298 
•03306 
•03313 
•03321 


•03637 
•03645 
•03653 
•03660 
.03668 


.03774 
.03783 
.03791 
•03799 
•03808 


30 

31 
32 
33 
34 


3u 
36 
37 
38 
39 


•02402 
•02408 
.02415 
-02421 
•02427 


.02461 
.02468 
•02474 
•02481 
•02488 


•02797 
•02804 
.02811 
.02818 
-02824 


.02878 
.02885 
.02892 
.02899 
.02907 


.03222 
.03229 
.03236 
.03244 
.03251 


.03329 
.03337 
.03345 
.03353 
03360 


•03676 
.03684 
.03692 
.03699 
•03707 


•03816 
•03825 
.03833 
.03842 
•03850 


35 
36 
37 
38 
39 


40 

41 
42 
43 

44 


•02434 
•02440 
.02447 
.02453 
•02459 


•02494 
•02501 
.02508 
.02515 
.02521 


•02831 
•02838 
.02845 
02852 
.02859 


.02914 
.02921 
.02928 
.02936 
.02943 


.03258 
.03266 
•03273 
•03281 
•03288 


•03368 
•03376 
.03384 
•03392 
.03400 


■03715 
.03723 
.03731 
.03739 
.03747 


.03858 
.03867 
.03875 
.03884 
.03892 


40 

41 
42 
43 
44 


45 
46 
47 

Us 

149 


•02466 
•02472 
.02479 
.02485 
.02492 


.02528 
.02535 
.02542 
.02548 
•02555 


.02866 
.02873 
.02880 
.02887 
.02894 


.02950 
.02958 
.02965 
.02972 
.02980 


•03295 
•03303 
•03310 
•03318 
.03325 


•03408 
•03416 
.03424 
•03432 
•03439 


.03754 
.03762 
03770 
.03778 
.03786 


.03901 
.03909 
.03918 
.03927 
.03935 


45 
46 
47 
48 

49 


50 

|51 
1 52 
1153 
^54 

55 
56 

157 
58 

'59 


•02498 
•02504 
•02511 
•02517 
•02524 


•02562 
.02569 
.02576 
.02582 
.02589 

•02596 
.02603 
.02610 
.02617 
•02624 


.02900 
•02907 
.02914 
.02921 
•02928 


.02987 
.02994 
.03002 
.03009 
.03017 


•03333 
.03340 
.03347 
.03355 
•03362 


•03447 
•03455 
.03463 
•03471 
•03479 


.03794 
.03802 
•03810 
.03818 
.03826 


.03944 
.03952 
.03961 
.03969 
.03978 


50 

51 

52 
53 
54 


.02530 
•02537 
.02543 
•02550 
.02556 


•02935 
•02942 
■02949 
•02956 
•02963 


.03024 
.03032 
.03039 
03046 
.03054 


•03370 
•03377 
•03385 
•03392 
.03400 


•03487 
•03495 
•03503 
•03512 
.03520 


.03834 
.03842 
•03850 
.03858 
.03866 


•03987 
.03995 
.04004 
.04013 
.04021 


55 
56 
57 
58 
59 


'60 


•02563 -02630 


.02970 


.03061 


03407 


.03528 


•03874 


.04030 


60 










r. 


21 











TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 





16 


o 


ir 




18° 


- 


19° 








t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec 


f 



1 

2 
3 
4 




1 

3 

4 


.03874 
•03882 
.03890 
03898 
.03906 


.04030 
.04039 
.04047 
.04056 
.04065 


.04370 
.04378 
•04387 
.04395 
.04404 


.04569 
.04578 
.04588 
.04597 
.04606 


.04894 
.04903 
•04912 
.04921 
.04930 


.05146 
.05156 
.05166 
.05176 
.05186 


•05448 
•05458 
•05467 
•05477 
.05486 




05762 
05773 
05783 
05794 
05805 


5 
6 
7 
8 
9 


.03914 
.03922 
.03930 
.03938 
.03946 


.04073 
.04082 
.04091 
.04100 
.04108 


.04412 
.04421 
.04429 
.04438 
.04446 


.04616 
.04625 
04635 
.04644 
.04653 


.04939 
.04948 
.04957 
.04967 
.04976 


.05196 
■05206 
.05216 
.05226 
.05236 


•05496 
.05505 
.05515 
.05524 
.05534 




05815 
05826 
05836 
05847 
05858 


5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
16 
17 
.18 
19 


10 

11 

12 
13 
14 


.03954 
.03963 
.03971 
.03979 
.03987 


.04117 
04126 
.04135 
.04144 
.04152 


•04455 
.04464 
.04472 
.04481 
.04489 


.04663 
.04672 
.04682 
.04691 
.04700 


.04985 
.04904 
.05003 
.05012 
.05021 


.05246 
.05256 
.05266 
.05276 
.05286 


.05543 
.05553 
.05562 
.05572 
.05582 




05869 
05879 
05890 
05901 
05911 


15 
16 
17 
18 
19 


.03995 
.04003 
.04011 
.04019 
.04028 


.04161 
.04170 
.04179 
.04188 
.04197 


.04498 
.04507 
.04515 
.04524 
.04533 


.04710 
.04719 
.04729 
.04738 
.04748 


.05030 
.05039 
.05048 
.05057 
.05067 


.05297 
.05307 
.05317 
.05327 
.05337 


.05591 
.05601 
056]0 
.05620 
•05630 




05922 
05933 
05944 
05955 
05965 


20 

21 
22 
23 
24 


.04036 
.04044 
•04052 
04060 
.04069 


.04206 
.04214 
.04223 
.04232 
.04241 


.04541 
.04550 
.04559 
.04567 
.04576 


.04757 
.04767 
.04776 
.04786 
.04795 


.05076 
.05085 
.05094 
.05103 
.05112 


•05347 
•05357 
.05367 
.05378 
.05388 


•05639 
•05649 
•05658 
.05668 
•05678 




05976 
05987 
05998 
06009 
06020 


30 

21 
22 
23 
24 
25 
26 
27 
28 
29 


25 
26 
27 
28 
29 


.04077 
.04085 
.04093 
04102 
.04110 


.04250 
.04259 
.04268 
.04277 
.04286 


.04585 
.04593 
.04602 
.04611 
.04620 


.04805 
.04815 
.04824 
04834 
.04843 


.05122 
.05131 
.05140 
.05149 
•05158 


.05398 
.05408 
.05418 
.05429 
.05439 


.05687 
•05697 
05707 
•05716 
.05726 




06030 
06041 
06052 
06063 
06074 


30 

31 
32 
33 
34 


.04118 
.04126 
.04135 
04143 
.04151 


.04295 
.04304 
.04313 
.04322 
.04331 


.04628 
.04637 
.04646 
04655 
.04663 


.04853 
.04863 
.04872 
.04882 
.04891 


.05168 
.05177 
.05186 
.05195 
.05205 


.05449 
.05460 
.05470 
.05480 
.05490 


•05736 
•05746 
.05755 
•05765 
•05775 




06085 
•06096 
.06107 
•06118 
•06129 


30 

31 
32 
33 
34 

35 
36 
37 
38 
-39 
40 
41 
42 
43 
44 


35 

36 

37 

38 

3£L 

40 

41 

42 

43 

44 


.04159 
.04168 
.04176 
.04184 
•04193 

.04201 
.04209 
.04218 
.04226 
•04234 


.04340 
.04349 
.04358 
.04367 
.04376 

.04385 
.04394 
.04403 
.04413 
.04422 


.04672 
.04681 
.04690 
.04699 
.04707 
•04716 
.04725 
.04734 
.04743 
.04752 


.04901 
•04911 
04920 
•04930 
.04940 

•04950 
.04959 
•04969 
■04979 
.04989 


.05214 
.05223 
.05232 
.05242 
.05251 
.05260 
.01270 
.05279 
.05288 
.05298 


.05501 
.05511 
•05521 
.05532 
.05542 
•05552 
.05563 
•05573 
•05584 
•05594 


•05785 
•05794 
.05804 
.05814 
.05824 
•05833 
.05843 
•05853 
•05863 
.05873 


- 


•06140 
•06151 
•06162 
•06173 
•06184 
•06195 
•06206 
•06217 
•06228 
06239 


45 
46 
47 
48 
49 


.04243 
.04251 
04260 
•04268 
.04276 


.04431 
.04440 
.04449 
.04458 
.04468 


.04760 
•04769 
.04778 
•04787 
•04796 


.04998 
.05008 
.05018 
.05028 
.05038 


.05307 
.05316 
.05326 
.05335 
.05344 


•05604 
•05615 
.05625 
•05636 
•05646 


.05882 
.05892 
.05902 
•05912 
.05922 




06250 
•06261 
06272 
06283 
06295 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


04285 
.04293 
.04302 
.04310 
•04319 


.04477 
.04486 
.04495 
.04504 
.04514 


.04805 
.04814 
.04823 
.04832 
•04841 


•05047 
.05057 
.05067 
.05077 
.05087 


.05354 
.05363 
.05373 
.05382 
•05391 


•05657 
•05667 
•05678 
.05688 
•05699 


.05932 
.05942 
.05951 
.05961 
.05971 




06306 
06317 
06328 
06339 
06350 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•04327 
•04336 
.04344 
•04353 
.04361 


.04523 
.04532 
.04541 
.04551 
•04560 
.04569 


•04850 
•04858 
•04867 
•04876 
04885 


.05097 
.05107 
.05116 
.05126 
.05136 


.05401 
.05410 
.05420 
.05429 
.05439 


.05709 
.05720 
.05730 
.05741 
•05751 


•05981 
•05991 
•06001 
•06011 
•06021 




06362 
06373 
06384 
06395 
06407 


55 
56 
57 
58 
59 


60 


•04370 


.04894 


.05146 


.05448 


.05762 


•06031 




06418 


60 



722 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS, 
30° 31° 33° 33° 



/ 


Vers. 


Exc sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 

4 


.06031 
-06041 
.06051 
.06061 
.06071 


.06418 
•06429 
•06440 
•06452 
06463 


.06642 
.06652 
.06663 
.06673 
.06684 


.07115 
.07126 
.07138 
.07150 
.07162 


.07282 
•07293 
•07303 
.07314 
.07325 


.07853 
.07866 
.07879 
.07892 
.07904 


.07950 
•07961 
.07972 
.07984 
.07995 


•08636 
•08649 
•08663 
•08676 
•08690 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.06081 
.06091 
.06101 
.06111 
06121 


•06474 
•06486 
06497 
.06508 
.06520 


.06694 
.06705 
.06715 
.06726 
.06736 


.07174 
.07186 
.07199 
.07211 
.07223 


.07336 
.07347 
07358 
•07369 
•07380 


.07917 
07930 
•07943 
.07955 
•07968 


.08006 
.08018 
.08029 
.08041 
-08052 


•08703 
•08717 
•08730 
•08744 
•08757 


5 
6 
7 
8 
9 


10 

11 
12 
13 

14 


.06131 
.06141 
•06151 
.06161 
.06171 


.06531 
.06542 
.06554 
•06565 
.06577 


.06747 
.06757 
.06768 
•06778 
.06789 


.07235 
•07247 
•07259 
.07271 
•07283 


•07391 
•07402 
.07413 
.07424 
•07435 


•07981 
.07994 
.08006 
.08019 
.08032 


.08064 
.08075 
.08086 
.08098 
.08109 


•08771 
•08784 
.08798 
.08811 
.08825 
.08839 
•08852 
.08866 
.08880 
•08893 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.06181 
.06191 
.06201 
.06211 
.06221 


•06588 
•06600 
.06611 
.06622 
.06634 


.06799 
.06810 
.06820 
.06831 
.06841 


•07295 
.07307 
.07320 
.07332 
•07344 


•07446 
.07457 
.07468 
.07479 
•07490 


.08045 
.08058 
.08071 
•08084 
.08097 


•08121 
.08132 
.08144 
•08155 
08167 


15 
13 
17 
18 


30 

21 
22 
23 
24 


.06231 
.06241 
.06252 
.06262 
•06272 


•06645 
.06657 
.06668 
.06680 
.06691 


.06852 
.06863 
.06873 
.06884 
.06894 


.07356 
.07368 
.07380 
.07393 
.07405 


•07501 
.07512 
•07523 
•07534 
•07545 


.08109 
.08122 
.08135 
.08148 
.08161 


.08178 
.08190 
•08201 
•08213 
•08225 


.08907 
.08921 
08934 
.08948 
.08962 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


■06282 
.06292 
.06302 
.06312 
.06323 


.06703 
.06715 
.06726 
.06738 
.06749 


.06905 
.06916 
•06926 
•06937 
.06948 


.07417 
.07429 
.07442 
.07454 
•07466 


•07556 
•07568 
•07579 
•07590 
•07601 


.08174 
.08087 
.08200 
.08213 
.08226 


•08236 
08248 
•08259 
•08271 
08282 


.08975 
.08989 
.09003 
.09017 
.09030 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.08333 
.06343 
.06353 
.06363 
.06374 


•06761 
.06773 
.06784 
.06796 
.06807 


.06958 
.06969 
.06980 
•06990 
•07001 


•07479 
.07491 
.07503 
.075L6 
.07528 


•07612 
.07623 
•07634 
•07645 
.07657 


.08239 
.08252 
.08265 
.08278 
.08291 


•08294 
•08306 
•08317 
•08329 
•08340 


.09044 
.09058 
.09072 
.09086 
.09099 


30 

31 
32 
38 
34 


35 
36 
37 
38 
39 


.06384 
.06394 
.06404 
.06415 
.06425 


.06819 
•06831 
•06843 
•06854 
•06866 


•07012 
•07022 
•07033 
. 07044 
.07055 


.07540 
.07553 
.07565 
.07578 
•07590 


•07668 
•07679 
.07690 
-07701 
.07713 


•08305 
.08318 
.08331 
.08344 
.08357 


•08352 
•08364 
.08375 
•08387 
.08399 


.09113 
•09127 
.09141 
.09155 
.09169 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


.06435 
.06445 
.06456 
.06466 
.06476 


.06878 
.06889 
06901 
.06913 
.06925 


.07065 
.07076 
.07087 
.07098 
.07108 


•07302 
.07615 
•07627 
•07640 
.07652 


•07724 
.07735 
.07746 
.07757 
•07769 


.08370 
.08383 
08397 
.08410 
•08423 


•08410 
•08422 
08434 
.08445 
•08457 


.09183 
.09197 
.09211 
.09224 
.09238 


40 

41 
42 
48 
44 


,45 
146 
147 
48 
|49 


06486 
.06497 
.06507 
.06517 
.06528 


.06936 
.06948 
.06960 
.06972 
-06984 


•07119 
•07130 
•07141 
.07151 
.07162 


.07665 
.07677 
.07690 
.07702 
•07715 


.07780 
•07791 
•07802 
07814 
•07825 


.08436 
.08449 
.08463 
.08476 
•08489 


08469 
.08481 
.08492 
.08504 
.08516 


.09252 
.09266 
.09280 
.09294 
09308 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.06538 
.06548 
.06559 
06569 
•06580 


.06995 
.07007 
.07019 
•07031 
.07043 


•07173 
.07184 
.07195 
.07206 
•07216 


.07727 
.07740 
.07752 
.07765 
.07778 


•07836 
07848 
07859 
•07870 
•07881 


.08503 
.08516 
.08529 
.08542 
.08556 


.08528 
•08539 
.08551 
.08563 
•08575 


.09323 
.09337 
.09351 
.09365 
09379 


50 

51 

52 
53 
54 


55 
56 
57 
58 
,59 


.06590 
.06600 
06611 
•06621 
•06632 


.07055 
•07067 
.07079 
.07091 
.07103 


.07227 
.07238 
• 07249 
.07260 
.07271 


.07790 
•07803 
.07816 
•07828 
•07841 


.07893 
•07904 
•07915 
•07927 
•07938 


.08569 
•08582 
•08596 
•08069 
•08623 


•08586 
•08598 
■08610 
•08622 
08634 


•09393 
.09407 
-09421 
•09435 
■09449 


55 
56 
57 
58 
59 


r 


06642 


.07115 


.07282 


•07853 


•07950 


•08636 


•08645 


•09464 


60 



723 



TABLE X.—NATURAL VERSED SINES AND EXTERNAL 
34° 25° 36° 37" 


SECANTa 


' 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 
4 


08645 
.08657 
.08669 
.08681 
.08693 


.09464 
.09478 
.09492 
.09506 
•09520 
.09535 
.09549 
.09563 
.09577 
.09592 


.09369 
.09382 
.09394 
.09406 
09418 




10338 
10353 
10368 
10383 
10398 


■10121 
.10133 
.10146 
.10159 
•10172 


•11260 
.11276 
.11292 
.11308 
.11323 


•10899 
•10913 
•10926 
•10939 
•10952 




12233 
1224b 
12266 
12283 
12299 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.08705 
.08717 
.08728 
.08740 
.08752 


•09431 
.09443 
.09455 
09468 
.09480 




10413 
10428 
10443 
10458 
10473 


•10184 
.10197 
.10210 
.10223 
.10236 


.11339 
11355 
.11371 
.11387 
•11403 


•10965 
•10979 
•10992 
•11005 
•11019 




12316 
12333 
12349 
12366 
12383 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.08764 
.08776 
.08788 
.08800 
.08812 


.09606 
.09620 
.09635 
.09649 
.09663 


.09493 
.09505 
•09517 
.09530 
.09542 




10488 
10503 
10518 
10533 
10549 


.10248 
.10261 
.10274 
.10287 
.10300 


.11419 
.11435 
.11451 
•11467 
•11483 


•11032 
•11045 
.11058 
11072 
.11085 




.12400 
.12416 
.12433 
.12450 
.12467 


10 

11 
12 
13 
14 
15 
16 
17 
18 
19 


15 
16 
17 
18 

19 


.08824 
.08836 
•08848 
.08860 
08872 


.09678 
.09o92. 
.09707 
.09721 
.09735 


09554 
.09567 
•09579 
•09592 
.09604 
.09617 
.09629 
•09642 
•09654 
.09666 


- 


10564 
10579 
10594 
10609 
10625 

10640 
10655 
10670 
10686 
10701 


•10313 
.10326 
•10338 
•10351 
•10364 
.10377 
.10390 
.10403 
.10416 
10429 


.11499 
11515 
.11531 
.11547 
•11563 


.11098 
.11112 
.11125 
•11138 
•11152 




.12484 
.12501 
.12518 
.12534 
•12551 


30 

21 
22 
23 
24 


.08884 
•08896 
•08908 
•08920 
•08932 


.09750 
.09764 
.09779 
.09793 
.09808 


.11579 
.11595 
•11611 
.11627 
.11643 


•11165 
•11178 
.11192 
.11205 
•11218 




.12568 
12585 

.12602 
12619 
12636 


30 

21 
22 
23 
24 
25 
26 
27 
28 
29 


25 
26 
27 
28 
29 


•08944 
•08956 
•08968 
•08980 
•08992 


.09822 
.09837 
.09851 
.09866 
.09880 


.09679 
.09691 
.09704 
.09716 
.09729 




10716 
10731 
10747 
10762 
10777 


.10442 
•10455 
•10468 
•10481 
•10494 


.11659 
.11675 
.11691 
.11708 
.11724 


.11232 
.11245 
.11259 
•11272 
•11285 




.12653 
12670 
12687 
12704 
12721 


30 

31 
32 
33 
34 


•09004 
•09016 
.03028 
.09040 
•09052 


.09895 
•09909 
•09924 
•09939 
•09953 


.09741 
.09754 
.09767 
.09779 
•09792 




1079a 
10808 
10824 
10839 
10854 


.10507 
.10520 
.10533 
•10546 
•10559 


.11740 
.11756 
.11772 
.11789 
.11805 


•11299 
.11312 
•11326 
.11339 
.11353 




12738 
12755 
12772 
12789 
12807 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.09064 
.09076 
.09089 
.09101 
•09113 


.09968 
•09982 
•09997 
.10012 
•10026 


•09804 
09817 
09829 
.09842 
•09854 




10870 
10885 
10901 
10916 
10932 


•10572 
•10585 
•10598 
•10611 
•10624 


.11821 
.11838 
.11854 
.11870 
•11886 


.11366 
.11380 
.11393 
.11407 
.11420 




12824 
12841 
12858 
12875 
12892 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•09125 
•09137 
•09149 
•09161 
.09174 


.10041 
.10055 
.10071 
.10085 
.10100 


•09867 
•09880 
.09892 
•09905 
•09918 




10947 
10963 
10978 
10994 
11009 


•10637 
.10650 
•10663 
•10676 
.10689 


•11903 
•11919 
.11936 
.11952 
•11968 


•11434 
•11447 
•11461 
•11474 
•11488 




12910 
12927 
12944 
12961 
12979 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•09186 
.09198 
.09210 
•09222 
•09234 


.10115 
.10130 
.10144 
.10159 
.10174 


•09930 
•09943 
•09955 
•09968 
.09981 




11C25 
11041 
11056 
11072 
11087 


•10702 
•10715 
•10728 
•10741 
•10755 


•11985 
•12001 
•12018 
•12034 
.12051 


•11501 
•11515 
-11528 
•11542 
•11555 




12996 
13013 
13031 
13048 
13065 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•09247 
.09259 
09271 
•09283 
•09296 


.10189 
.10204 
.10218 
.10233 
.10248 


•09993 
•10006 
•10019 
•10032 
•10044 




11103 
11119 
11134 
11150 
11166 


•10768 
•10781 

10794 
•10807 

10820 


.12067 
•12084 
.12100 
•12117 
•12133 


•11569 

•11583 

11596 

11610 

•11623 




13083 
13100 
13117 
13135 
13152 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•09308 
•09320 
•09332 
09345 
•09357 


.10263 
.10278 
.10293 
.10308 
.10323 

.10338 


•10057 
•10070 
•10082 
•10095 
-10108 
•10121 




11181 
11197 
11213 
11229 
11244* 


•10833 
•10847 
•10860 
•10873 
•10886 


•12150 
.12166 
.12183 
.12199 
.12216 


•11637 
•11651 
.11664 
.11678 
.11692 




13170 
13187 
13205 
13222 
13240 


55 
56 
57 
58 
59 


60 


•09369 




11260 


•10899 


.12233 


•11705 




13257 


60 












72 


4 













2TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
) 28° 39° 30° 31° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


t 


L 

1 1 
S 2 

: 3 

f 4 


.11705 
.11719 
.11733 
.11746 
•11760 


.13257 
.13275 
.13292 
.13310 
.13327 


.12538 
.12552 
.12566 
.12580 
.12595 




14335 
14354 
14372 
14391 
14409 


•13397 
.13412 
.13427 
•13441 
•13456 




15470 
15489 
15509 
15528 
15548 


. 14283 
.14298 
.14313 
.14328 
. 14343 




16663 
.16684 
.16704 
.16725 
•16745 




1 
2 
3 
4 


5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 
16 
17 
18 
19 


.11774 
.11787 
•11801 
•11815 
•11828 
•11842 
.11856 
.11870 
.11883 
•11897 


.13345 
•13362 
.13380 
.13398 
.13415 


.12609 
.12623 
.12637 
.12651 
.12665 




14428 
14446 
14465 
14483 
14502 


.13470 
.13485 
.13499 
•13514 
.13529 




•15567 
•15587 
•15606 
•15626 
15645 


.14358 
.14373 
.14388 
. 14403 
•14418 




.16766 
.16786 
.16806 
.16827 
.16848 


5 
6 
7 
8 
9 


.13433 
•13451 
.13468 
.13486 
.13504 


.12679 
.12694 
.12708 
.12722 
.12736 




14521 
14539 
.14558 
.14576 
14595 


.13543 
.13558 
.13573 
.13587 
.13602 




.15665 
•15684 
•15704 
•15724 
•15743 


. 14433 
. 14449 
. 14464 
. 14479 
. 14494 




.16868 
.16889 
.16909 
.16930 
•16950 


10 

11 

12 
13 
14 


.11911 
•11925 
.11938 
.11952 
.11966 


•13521 
.13539 
.13557 
.13575 
.13593 


.12750 
.12765 
.12779 
.12793 
.12807 




.14614 
.14632 
.14651 
•14670 
14689 


.13616 
.13631 
.13646 
•13660 
•13675 




•15763 
•15782 
•15802 
.15822 
•15841. 


.14509 
.14524 
.14539 
.14554 
.14569 




•16971 
•16992 
•17012 
•17033 
.17054 


15 
16 
17 
18 

JL9 


20 

21 
22 
23 
24 


.11980 
.11994 
.12007 
.12021 
.12035 


.13610 
.13628 
.13646 
.13664 
.13682 


.12822 
.12836 
.12850 
.12864 
.12879 




.14707 
.14726 
.14745 
•14764 
14782 


•13690 
.13705 
.13719 
.13734 
.13749 




•15861 
.15881 
.15901 
.15920 
•15940 


.14584 
.14599 
.14615 
.14630 
.14645 




.17075 
•17095 
•17116 
•17137 
•17158 


20 

21 
22 
23 

24 


25 
26 
27 
28 
29 


.12049 
.12063 
.12077 
.12091 
•12104 


•13700 
.13718 
.13735 
.13753 
.13771 


.12893 
.12907 
.12921 
.12936 
.12950 




.14801 
.14820 
•14839 
14858 
14877 


.13763 
•13778 
.13793 
.13808 
.13822 




•15960 
•15980 
•16000 
16019 
16039 


.14660 
.14675 
.14690 
•14706 
•14721 




•17178 
.17199 
.17220 
•17241 
17262 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.12118 

.12132 

•12146 

12160 

12174 


.13789 
.13807 
.13825 
.13843 
.13861 


.12964 
.12979 
.12993 
.13007 
.13022 




14896 
14914 
14933 
14952 
14971 


.]3837 
.13852 
.13867 
•13881 
.13896 




16059 
16079 
16099 
16119 
16139 


•14736 
•14751 
•14766 
•14782 
•14797 




17283 
17304 
17325 
17346 
17367 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.12188 
.12202 
.12216 
•12230 
.12244 


.13879 
.13897 
.13916 
.13934 
13952 


.13036 
.13051 
.13065 
.13079 
.13094 




14990 
15009 
15028 
15047 
15066 


.13911 
.13926 
.13941 
.13955 
.18970 




16159 
16179 
16199 
16219 
16239 


•14812 
•14827 
•14843 
•14858 
.14873 




17388 
17409 
17430 
17451 
17472 


35 
36 
37 
38 
_39 


40 

41 
42 
43 
44 


.12257 
.12271 
.12285 
.12299 
.12313 


•13970 
.13988 
.14006 
.14024 
.14042 


.13108 
.13122 
.13137 
.13151 
.13166 




15085 
1510J 
15124 
15143 
15162 


13985 
• 14000 
•140] 5 
. 14030 
.14044 




16259 
16279 
16299 
16319 
16339 


•14888 
•14904 
•]4ai9 
•14934 
.14949 




17493 
17514 
17535 
17556 
17^77 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


.12327 
.12341 
.12355 
.12369 
12383 


.14061 
.14079 
.14097 
.14115 
.14134 


.13180 
13195 
.13209 
.13223 
.13238 




15181 
15200 
15219 
15239 
15258 


- 14059 
14074 
.14089 
.14104 
•14119 




16359 
16380 
16400 
16420 
16440 


•14965 
•14980 
.14995 
•15011 
.15026 




17598 
17620 
17641 
17662 
1768-3 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.12397 
.12411 
.12425 
. 12439 
.12454 


.14152 
.14170 
.14188 
.14207 
.14225 


.13252 
.13267 
.13281 
.13296 
.13310 




15277 
15296 
15315 
15335 
15354 


.14134 
.14149 
.14164 
.14179 
.14194 

.14208 
.14223 
.14238 
.14253 
.14268 

.14283 


- 


16460 
16481 
16501 
16521 
16541 

16562 
16582 
16602 
16623 
16643 
16663 


15041 
.15057 
.15072 
.15087 

15103 

15118 
.15134 
.15149 
.15164 
-15180 

.15195 


- 


17704 
17726 
17747 
17768 
17790 

17811 
17832 
17854 
17875 
17896 
17918 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 

60 


.12468 
.12482 
.12496 
.12510 
.12524 
.12538 


.14243 
14262 
.14280 
.14299 
.14317 
.14335 


•13325 
.13339 
.13354 
.13368 
.13383 
.13397 


- 


15373 
15393 
15412 
15431 
15451 
15470 


55 
56 
57 
58 
59 
60 



725 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
) 33° 33° 34° 35° ^ 



t 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


# 




1 

2 
3 
4 


.15195 
.15211 
.15226 
.15241 
.15257 




17918 

17939 

.17961 

.17982 

.18004 


.16133 
.16149 
.16165 
.16181 
.16196 




.19236 
.19259 
.19281 
.19304 
.19327 


.17096 
.17113 
.17129 
.17145 
.17161 




.20622 
.20645 
.20669 
.20693 
.20717 


.18085 
.18101 
.18118 
.18135 
.18152 


.22077 
.22102 
.22127 
.22152 
.22177 




1 

2 
3 

4 


5 

f 

8 

9 


.15272 
.15288 
.15303 
.15319 
.15334 




.18025 
.18047 
.18068 
.18090 
18111 


.16212 
.16228 
.16244 
.16260 
.16276 




.19349 
.19372 
.19394 
.19417 
. 19440 


.17178 
.17194 
.17210 
.17227 
.17243 




-20740 
.20764 
.20788 
.20812 
.20836 


.18168 
.18185 
.18202 
.18218 
.18235 


.22202 
.22227 
.22252 
.22277 
.22302 


3 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.15350 
.15365 
.15381 
.15396 
.15412 




18133 
18155 
18176 
18198 
18220 


.16292 
.16308 
.16324 
.16340 
.16355 




.19463 
.19485 
.19508 
.19531 
.19554 


.17259 
.17276 
.17292 
.17308 
.17325 




.20859 
.20883 
.20907 
.20931 
.20955 


.18252 
.18269 
.18286 
.18302 
.18319 


.22327 
.22352 
.22377 
. 22402 
.22428 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.15427 
.15443 
.15458 
.15474 
.15489 




18241 
18263 
18285 
18307 
18328 


.16371 
.16387 
.16403 
.16419 
.16435 




.19576 
.19599 
.19622 
.19645 
.19668 


.17341 
.17357 
.17374 
.17390 
.17407 




.20979 
.21003 
•21027 
.21051 
.21075 


.18336 
.18353 
.18369 
.18386 
.18403 


.22453 
.22478 
.22503 
.22528 
.22554 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


.15505 
.15520 
.15536 
.15552 
.15567 




18350 
18372 
18394 
18416 
18437 


.16451 
.16467 
.16483 
.16499 
•16515 




19691 

19713 

19736 

.19759 

.19782 


.17423 
.17439 
.17456 
.17472 
.17489 




.21099 
.21123 
.21147 
.21171 
.21195 


.18420 
.18437 
.18454 
.18470 
.18487 


.22579 
.22604 
.22629 
.22655 
.22680 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.15583 
.15598 
.15614 
.15630 
.15645 




18459 
18481 
18503 
18525 
18547 


•16531 
.16547 
•16563 
.16579 
16595 




19805 
19828 
19851 
19874 
19897 


.17505 
.17522 
.17538 
.17554 
.17571 




.21220 
.21244 
.21268 
.21292 
.21316 


.18504 
.18521 
.18538 
.18555 
.18572 


.22706 
.22731 
.22756 
.22782 
.22807 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.15661 
.15676 
.15692 
.15708 
.15723 




18569 
18591 
18613 
18635 
18657 


.16611 
.16627 
.16644 
.16660 
•16676 




19920 
19944 
19967 
19990 
20013 


.17587 
.17604 
.17620 
.17637 
.17653 




.21341 
.21365 
21389 
21414 
21438 


.18588 
.18605 
.18622 
.18639 
.18656 


.22833 
.22858 
.22884 
.22909 
.22935 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.15739 
.15755 
.15770 
.15786 
.15802 




18679 
18701 
18723 
18745 
18767 


.16692 
.16708 
.16724 
.16740 
•16756 




20036 
20059 
20083 
20106 
20129 


.17670 
.17686 
.17703 
.17719 
•17736 




21462 
21487 
21511 
21535 
21560 


.18673 
.18690 
.18707 
. 18724 
•18741 


.22960 
.22986 
.23012 
.23037 
.23063 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


.15818 
.15833 
.15849 
.15865 
.15880 




18790 
18812 
18834 
18856 
18878 


.16772 
.16788 
.16805 
.16821 
.16837 




20152 
20176 
20199 
20222 
20246 


•17752 
.17769 
.17786 
•17802 
•17819 




21584 
21609 
21633 
21658 
21682 


•18758 
•18775 
•18792 
•18809 
.18826 


.23089 
.23114 
.23140 
.23166 
.23192 


40 

41 

42 
43 
44 


45 
46 
47 
48 
49 


.15896 
.15912 
.15928 
.15943 
•15959 




18901 
18923 
18945 
18967 
18990 


•16853 
.16869 
.16885 
.16902 
.16918 




20269 
20292 
20316 
20339 
20363 


•17835 
.17852 
•17868 
•17885 
•17902 




21707 
21731 
21756 
21781 
21805 


•18843 
•18860 
•18877 
.18894 
•18911 


.23217 
.23243 
.23269 
.23295 
.23321 


45 
46 
47 
48 
49 


60 

51 
52 
53 
54 


.15975 
.15991 
.16006 
.16022 
.18038 




19012 
19034 
19057 
19079 
19102 


•16934 
.16950 
•16966 
•16983 
•16999 




20386 
20410 
20433 
20457 
20480 


.17918 
.17935 
.17952 
•17968 
•17985 




21830 
21855 
21879 
21904 
21929 


•18928 
•18945 
•18962 
.18979 
•18996 


.23347 
.23373 
.23399 
.23424 
•23450 


50 

51 

52 
53 
54 


55 
56 
57 
58 
59 


.16054 
.16070 
.16085 
•16101 
•16117 




19124 
19146 
19169 
19191 
19214 


•17015 
•17031 
•17047 
•17064 
•17080 




20504 
20527 
20551 
20575 
20598 


•18001 
•18018 
•18035 
•18051 
•18068 




21953 
21978 
22003 
22028 
22053 


•19013 

.19030 

.19047 

19064 

19081 


.23476 
.23502 
.23529 
.23555 
•23581 


55 
56 
57 
58 
59 


60 


.16133 




19236 


.17096 




20622 


.18085 




22077 


19098 


.23607 


60 



726 



\ 

IWBLE X.— NATURAL VERSED SINES AND EXTERNAL 
36° 37° 38° 39' 


SECANTS 


f 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 







1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

u 

12 
13 
14 
15 
16 
17 
18 
19 


•19098 
.19115 
•19133 
.19150 
.19167 


.23607 
.23633 
.23659 
.23685 
.23711 


•20136 
•20154 
•20171 
.20189 
.20207 
.20224 
.20242 
.20259 
.20277 
•20294 




25214 
25241 
25269 
25296 
25324 


21199 
21217 
•21235 
•21253 
•21271 




26902 
26931 
26960 
26988 
27017 


22285 
22304 

•22322 
22340 

•22359 




28676 
28706 
28737 
28767 
28797 




1 
2 

3 

4 


.19184 
.19201 
.19218 
.19235 
.19252 
.19270 
.19287 
.19304 
.19321 
.19338 


.23738 
.23764 
.23790 
.23816 
.23843 




25351 
25379 
25406 
25434 
25462 


.21289 
.21307 
.21324 
.21342 
•21360 




27046 
27075 
27104 
27133 
27162 


•22377 
•22395 
•22414 
•22432 
•22450 




28828 
28858 
28889 
28919 
28950 


5 
6 
7 
8 
9 


.23869 
.23895 
.23922 
.23948 
.23975 


.20312 
.20329 
•20347 
.20365 
.20382 


- 


25489 
25517 
25545 
25572 
25600 
25628 
25656 
25683 
25711 
25739 


•21378 
•21396 
•21414 
•21432 
•21450 




27191 
27221 
27250 
27279 
27308 


•22469 
•22487 
•22506 
•22524 
•22542 




28980 
29011 
29042 
29072 
29103 


10 

11 
12 
13 
14 


.19356 
.19373 
•19390 
•19407 
•19424 


.24001 
.24028 
.24054 
.24081 
•24107 


.20400 
.20417 
.20435 
.20453 
.20470 


•21468 
•21486 
•21504 
•21522 
•21540 




27337 
27366 
27396 
27425 
27454 


•22561 
•22579 
•22598 
•22616 
•22634 




29133 
29164 
29195 
29226 
29256 


15 
16 
17 
18 
19 


30 

i 21 

1 22 

1 23 

! 24 

25 

26 

* 27 

28 

1 29 


•19442 
•19459 
.19476 
.19493 
.19511 


.24134 
.24160 
.24187 
.24213 
.24240 


.20488 
.20506 
.20523 
•20541 
•20559 




25767 
25795 
25823 
25851 
25879 


•21558 
•21576 
•21595 
•21613 
•21631 




27483 
27513 
27542 
27572 
27601 


•22653 
•22671 
•22690 
•22708 
•22727 




29287 
29318 
29349 
29380 
29411 


30 

21 
22 
23 
24 


.19528 
.19545 
.19562 
.19580 
•19597 


.24267 
.24293 
.24320 
.24347 
.24373 


.20576 
.20594 
.20612 
.20629 
.20647 




25907 
25935 
25963 
25991 
26019 


•21649 
•21667 
•21685 
•21703 
•21721 




27630 
27660 
27689 
27719 
27748 


•22745 
•22764 
•22782 
•22801 
•22819 




29442 
29473 
29504 
29535 
29566 


25 
26 
27 
28 
29 


( 30 

i 31 

' 32 

33 

34 

35 
36 
37 
38 
39 


•19614 
•19632 
•19649 
•19666 
•19684 


.24400 
.24427 
.24454 
.24481 
.24508 


•20665 
•20682 
•20700 
•20718 
•20736 




26047 
260^5 
26104 
26132 
26160 


•21739 
•21757 
•21775 
•21794 
•21812 




27778 
27807 
27837 
27867 
27896 


•22838 
.22856 
•22875 
.22893 
•22912 




29597 
29628 
29659 
29690 
29721 


30 

31 

32 
33 
34 


•19701 
.19718 
.19736 
.19753 
•19770 


.24534 
.24561 
.24588 
.24615 
• 24642 


.20753 
•20771 
•20789 
.20807 
•20824 




26188 
26216 
26245 
26273 
26301 


•21830 
•21848 
•21866 
•21884 
•21902 




27926 
27956 
27985 
28015 
28045 


.22930 
.22949 
•22967 
•22986 
•23004 




29752 
29784 
29815 
29846 
29877 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


.19788 
.19805 
•19822 
•19840 
•19857 


.24669 
•24696 
•24723 
•24750 
•24777 


•20842 
-20860 
•20878 
•20895 
•20913 




26330 
26358 
26387 
26415 
26443 


•21921 
•21939 
•21957 
•21975 
•21993 




28075 
28105 
28134 
28164 
28194 


•23023 
•23041 
•23060 
•23079 
•23097 




29909 
29940 
29971 
30003 
30034 


40 
41 

42 
43 
44 


45 
46 
47 
48 
49 


•19875 
•19892 
.19909 
•19927 
•19944 


.24804 
.24832 
24859 
•24886 
•24913 


•20931 
•20949 
•20967 
•20985 
.21002 




26472 
26500 
26529 
26557 
26586 


•22012 
•22030 
•22048 
•22066 
•22084 

•22103 
•22121 
•22139 
•22157 
•22176 




28224 
.28254 
.28284 
.28314 
•28344 


•23116 
•23134 
•23153 
•23172 
•23190 




30066 
30097 
30129 
.30160 
.30192 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•19962 
•19979 
•19997 
•20014 
•20032 


.24940 
.24967 
.24995 
.25022 
.25049 


•21020 
•21038 
•21056 
•21074 
•21092 




26615 
26643 
26672 
26701 
.26729 




28374 
.28404 
.28434 
.28464 
•28495 


•23209 
•23228 
•23246 
•23265 
.23283 




.30223 
.30255 
.30287 
.30318 
.30350 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•20049 
•20066 
•20084 
•20101 
•20119 


.25077 
.25104 
.25131 
.25159 
•25186 


•21109 
•21127 
•21145 
•21163 
•21181 




.26758 
.26787 
.26815 
.26844 
•26873 


•22194 
•22212 
•22231 
•22249 
•22267 




.28525 
.28555 
.28585 
.28615 
•28646 


•23302 
•23321 
•23339 
•23358 
•23377 




.30382 
.30413 
.30445 
.30477 
•30509 


55 
56 
57 
56 

59 


60 


.20136 


.25214 


•21199 




.26902 


•22285 




.28676 


.23396 




.30541 


60 



727 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
40'' 41° 43*^ 43° 


/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


9 




1 
2 
3 

4 


•23396 
.23414 
.23433 
•23452 
•23470 


.30541 
.30573 
.30605 
.30636 
.30668 


.24529 
.24548 
.24567 
.24586 
.24605 


.32501 
.32535 
.32568 
.32602 
.32636 


.25686 
.25705 
.25724 
.25744 
.25763 


.34563 
.34599 
.34634 
.34669 
.34704 


.26865 
•26884 
•26904 
•26924 
•26944 


.36733 
.36770 
.36807 
.36844 
•36881 




1 

2 
3 

4 


5 
6 
7 
8 
9 


•23489 
•23508 
.23527 
.23545 
•23564 


.30700 
.30732 
.30764 
.30796 
.30829 


.24625 
.24644 
.24663 
.24682 
.24701 


.32669 
.32703 
.32737 
.32770 
.32804 


.25783 
.25802 
.25822 
•25841 
•25861 


.34740 
.34775 
.34811 
.34846 
•34882 


.26964 
.26984 
.27004 
•27024 
. 27043 


.36919 
.36956 
.36993 
.37030 
.37068 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.23583 
.23602 
.23620 
•23639 
•23658 


.30861 
.30893 
.30925 
.30957 
.30989 


.24720 
.24739 
.24759 
.24778 
.24797 


.32838 
.32872 
.32905 
.32939 
.32973 


.25880 
•25900 
.25920 
.25939 
•25959 


.34917 
.34953 
.34988 
.35024 
.35060 


« 27063 
.27083 
•27103 
•27123 
.27143 


.37105 
.37143 
.37180 
.37218 
.37255 


10 

11 
12 
13 
14 

15 
16 
17 
18 
19 


15 
16 
17 
18 
19 


.23677 
•23696 
•23714 
.23733 
.23752 


.31022 
.31054 
.31086 
.31119 
.31151 


.24816 
.24835 
.24854 
.24874 
.24893 


.33007 
.33041 
.33075 
.33109 
•33143 


.25978 
.25998 
.26017 
.26037 
•26056 


.35095 
.35131 
.35167 
.35203 
.35238 


.27163 
.27183 
•27203 
.27223 
.27243 


.37293 
.37330 
.37368 
.37406 
•37443 


20 

21 
22 
23 
24 


•23771 
•23790 
•23808 
•23827 
•23846 

•23865 
•23884 
•23903 
•23922 
•23941 


.31183 
.31216 
.31248 
.31281 
.31313 


•24912 
.24931 
.24950 
.24970 
.24989 


.33177 
.33211 
.33245 
.33279 
.33314 


.26076 
.26096 
.26115 
•26135 
•26154 


.35274 
.35310 
.35346 
.35382 
.35418 


.27263 
.27283 
.27303 
.27323 
.27343 


.37481 
.37519 
.37556 
.37594 
•37632 


30 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 


25 
26 
27 
28 
29 


.31346 
.31378 

.31411 
.31443 
.31476 


•25008 
•25027 
.25047 
.25066 
.25085 


.33348 
.33382 
.33416 
.33451 
•33485 


.26174 
.26194 
.26213 
.26233 
.26253 


.35454 
.35490 
.35526 
.35562 
•35598 


•27363 
.27383 
.27403 
•27423 
•27443 


.37670 
.37708 
.37746 
.37784 
.37822 


30 

31 
32 
33 
34 


•23959 
•23978 
•23997 
•24016 
•24035 


.31509 
.31541 
.31574 
.31607 
.31640 


•25104 
.25124 
.25143 
.25162 
•25182 


.33519 
.33554 
.33588 
.33622 
.33657 


.26272 
.26292 
•26312 
.26331 
•26351 


.35634 
.35670 
.35707 
.35743 
•35779 


.27463 
•27483 
.27503 
.27523 
.27543 


.37860 
.37898 
.37936 
.37974 
.38012 


35 
36 
37 
38 
39 


•24054 
•24073 
•24092 
•24111 
.24130 


.31672 
.31705 
.31738 
.31771 
.31804 


.25201 
.25220 
.25240 
.25259 
•25278 


.33691 
.33726 
.33760 
.33795 
•33830 


•26371 
•26390 
•26410 
•26430 
.26449 


.35815 
.35852 
.35888 
.35924 
.35961 


•27563 
.27583 
.27603 
.27623 
•27643 


.38051 
.38089 
.38127 
.38165 
•38204 


35 
36 
37 
38 
-39 
40 
41 
42 
43 
44 


40 

41 
42 
43 
44 


•24149 
.24168 
.24187 
.24206 
•24225 


.31837 
.31870 
.31903 
.31936 
.31969 


•25297 
.25317 
.25336 
.25356 
.25375 


.33864 
.33899 
.33934 
.33968 
•34003 


.26469 
•26489 
•26509 
•26528 
.26548 


.35997 
.36034 
.36070 
.36107 
.36143 


.27663 
•27683 
.27703 
.27723 
•27743 


.38242 
.38280 
.38319 
.38357 
•38396 


45 
46 
47 
48 
49 


.24244 
•24262 
.24281 
.24300 
.24320 


.32002 
.32035 
.32068 
.32101 
.32134 


.25394 
.25414 
.25433 
.25452 
.25472 


.34038 
.34073 
.34108 
.34142 
•34177 


•26568 
.26588 
.26607 
.26627 
•26647 


.36180 
.36217 
.36253 
.36290 
•36327 


•27764 
•27784 
•27804 
.27824 
.27844 


.38434 
.38473 
.38512 
.38550 
.38589 


45 
46 
47 
48 
49 


60 

51 
52 
53 
54 


.24339 
.24358 
•24377 
•24396 
•24415 


.32168 
.32201 
.32234 
.32267 
.32301 


•25491 
.25511 
•25530 
•25549 
•25569 


.34212 
.34247 
.34282 
.34317 
•34352 


.26667 
•26686 
•26706 
.26726 
.26746 


.36363 
.36400 
•36437 
.36474 
•36511 


•27864 
•27884 
.27905 
.27925 
.27945 


•38628 
.38666 
.38705 
.38744 
•38783 


50 

51 
52 
53 
54 


55 
56 
57 
58 

59 


•24434 
•24453 
.24472 
•24491 
.24510 


.32334 
.32368 
.32401 
.32434 
•32468 


.25588 
.25608 
.25627 
•25647 
.25666 


.34387 

.34423 
.34458 
.34493 
•34528 


.26766 
.26785 
•26805 
•26825 
•26845 


.36548 
.36585 
.36622 
.36669 
•36696 


.27965 
•27985 
•28005 
.28026 
•28046 


•38822 
.38860 
•38899 
.38938 
.38977 


55 
56 
57 
58 
-59 


60 


.24529 


.32501 


•25686 


•34563 


•26865 


.36733 


•28066 


.39016 


60 



728 



I'fABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
\ 44° 45° 46° 47** 



i 


Vers. 


Ex, sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 


. 


.28066 


.39016 


•29289 


.41421 


•30534 


.43956 


.31800 


•46628 


^'"o 


1 


.28086 


.39055 


.29310 


.41463 


•30555 


.43999 


•31821 


.46674 


1 

2 
3 
4 


: 2 


.28106 


.39095 


■29330 


.41504. 


.30576 


.44042 


•31843 


.46719 


3 


.28127 


.39134 


.29351 


.41545 


.30597 


.44086 


•31864 


•46765 


4 


.28147 


.39173 


.29372 


.41586 


.30618 


•44129 


•31885 


•46811 


5 


.28167 


.39212 


.29392 


.41627 


.30639 


•44173 


.31907 


•46857 


5 
6 
7 
8 
9 


6 


.28187 


.39251 


.29413 


.41669 


.30660 


.44217 


•31928 


•46903 
•46949 
•46995 
■^7041 


7 


.28208 


.39291 


.29433 


.41710 


.30681 


.44260 


•31949 


8 


.28228 


.39330 


.29454 


•41752 


.30702 


.44304 


•31971 


9 


.28248 


.39369 


•29475 


.41793 


•30723 


.44347 


•31992 


10 


.28268 


.39409 


•29495 


.41835 


.30744 


.44391 


•32013 


•47087 
.47134 
•47180 


10 

11 
12 


11 


.28289 


.39448 


.29516 


.41876 


.30765 


.44435 


•32035 


12 


.28309 


.39487 


.29537 


.41918 


.30786 


.44479 


•32056 


13 


•28329 


.39527 


.29557 


.41959 


•30807 


.44523 


•32077 


.47226 


13 


14 


.28350 


.39566 


.29578 


•42001 


.30828 


.44567 


•32099 


•47272 


14 


15 


.28370 


.39606 


.29599 


.42042 


.30849 


•44610 


.32120 


.47319 


15 


16 


.28390 


.39646 


.29619 


.42084 


•30870 


.44654 


.32141 


.47365 


16 


17 


.28410 


.39685 


.29640 


.42126 


•30891 


.44698 


•32163 


.47411 


17 


18 


.28431 


.39725 


.29661 


.42168 


.30912 


.44742 


•32184 


.47458 


18 


19 


.28451 


•39764 


•29681 


.42210 


•30933 


•44787 


•32205 


•47504 


19 


20 


.28471 


.39804 


•29702 


.42251 


•30954 


.44831 


•32227 


.47551 


20 


2i 


.28492 


.39844 


.29723 


.42293 


-30975 


.44875 


.32248 


•47598 


21 


22 


.28512 


.39884 


.29743 


.42335 


.30996 


.44919 


•32270 


.47644 


?.?. 


23 


•28532 


.39924 


.29764 


.42377 


.31017 


.44963 


.32291 


•47691 


28 


24 


•28553 


•39963 


.29785 


.42419 


•31038 


.45007 


.32312 


•47738 


24 


25 


•28573 


.40003 


.29805 


.42461 


•31059 


.45052 


•32334 


•47784 


25 


26 


.28593 


.40043 


•29826 


.42503 


.31080 


.45096 


•32355 


•47831 


26 


27 


.28614 


.40083 


•29847 


.42545 


•31101 


.45141 


•32377 


.47878 


27 


28 


.28634 


.40123 


•29868 


.42587 


.31122 


.45185 


.32398 


.47925 


28 


29 


.28655 


.40163 


.29888 


.42630 


.31143 


•45229 


•32420 


.47972 


29 


30 


•28675 


.40203 


•29909 


.42672 


.31165 


.45274 


•32441 


.48019 


30 


31 


•28695 


.40243 


•29930 


.42714 


•31186 


.45319 


.32462 


.48066 


31 


32 


•28716 


.40283 


•29951 


.42756 


•31207 


.45363 


•32484 


•48113 


32 


33 


.28736 


.40324 


.29971 


.42799 


.31228 


.45408 


•32505 


•48160 


33 


34 


•28757 


.40364 


.29992 


.42841 


•31249 


•45452 


•32527 


.48207 


34 


35 


•28777 


.40404 


.30013 


.42883 


•31270 


.45497 


•32548 


.48254 


35 


36 


•28797 


.40444 


.30034 


.42926 


.31291 


.45542 


•32570 


.48301 


36 


37 


•28818 


.40485 


.30054 


.42968 


.31312 


.45587 


.32591 


.48349 


37 


38 


.28838 


.40525 


•30075 


.43011 


•31334 


•45631 


•32613 


.48396 


38 


39 


.28859 


.40565 


•30096 
•30117 


•43053 
.43096 


■31355. 
.31376 


•45676 
.45721 


.32634 
•32656 


48443 
.48491 


39 


40 


•28879 


.40606 


40 


41 


•28900 


.40646 


•30138 


.43139 


•31397 


.45766 


•32677 


.48538 


41 


42 


•28920 


.40687 


•30158 


.43181 


.31418 


.45811 


•32699 


.48586 


42 


Aii 


.28941 


.40727 


.30179 


.43224 


.31439 


.45856 


•32720 


.48633 


43 


4:4: 


.28961 


.40768 


•30200 


•43267 


.31461 


•45901 


.32742 


.48681 


44 


45 


.28981 


.40808 


•30221 


.43310 


•31482 


.45946 


•32763 


•48728 


45 


46 


.29002 


.40849 


•30242 


.43352 


•31503 


.45992 


.32785 


.48776 


46 


47 


.29022 


.40890 


.30263 


.43395 


.31524 


.46037 


.32806 


•48824 


47 


48 


.29043 


.40930 


.30283 


.43438 


•31545 


-46082 


•32828 


.48871 


48 


49 


•29063 


.40971 


.30304 


.43481 


•31567 


-46127 


•32849 


48919 


49 


50 


.29084 


.41012 


.30325 


.43524 


.31588 


.46173 


•32871 


.48967 


50 


51 


•29104 


.41053 


•30346 


.43567 


•31609 


.46218 


•32893 


.49015 


51 


52 


•29125 


.41093 


.30367 


.43610 


.31630 


.46263 


•32914 


.49063 


52 


53 


.29145 


.41134 


•30388 


.43653 


.31651 


.46309 


•32936 


.49111 


53 


54 


.29166 


.41175 


.30409 


.43696 


•31673 


•46354 


32957 


•49159 


54 


55 


.29187 


.41216 


•30430 


.43739 


•31694 


.46400 


32979 


.49207 


55 


56 


•29207 


.41257 


•30451 


.43783 


•31715 


.46445 


33001 


.49255 


56 


bV 


•29228 


.41298 


.30471 


.43826 


•31736 


.46491 


33022 


.49303 


57 


58 


•29248 


.41339 


.30492 


.43869 


.31758 


.46537 


33044 


.49351 


58 


b9_ 


.29269 
.29289 


.41380 
.41421 


•30513 
•30534 


.43912 


31779 


.46582 _ 


33065 


.49399 


59 


60 


.43956 


31800 


.46628 • 


33087 


.49448 


60 



729 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
48° 49° 50° 51° 



} 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 

2 
3 

4 


•33087 
.33109 
.33130 
.33152 
.33173 


.49448 
.49496 
.49544 
.49593 
.49641 


.34394 
.34416 
.34438 
.34460 
.34482 


.52425 
.52476 
.52527 
.52579 
.52630 


•35721 
•35744 
.35766 
•35788 
•35810 


.55572 
.55626 
.55680 
.55734 
.55789 


•37068 
•37091 
.37113 
.37136 
.37158 


.58902 
.58959 
•59016 
.59073 
.59130 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.33195 
.33217 
.33238 
• 33260 
.33282 


.49690 
.49738 
.49787 
.49835 
.49884 


.34504 
•34526 
•34548 
•34570 
•34592 


.52681 
.52732 
.52784 
.52835 
•52886 

.52938 
.52989 
.53041 
.53092 
.53144 


•35833 
•35855 
•35877 
•35900 
•35922 


.55843 
.55897 
•55951 
.56005 
.56060 


•37181 
.37204 
.37226 
.37249 
.37272 


.59188 
.59245 
.59302 
.59360 
.59418 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.33303 
•33325 
.33347 
.33368 
.33390 


.49933 
.49981 
.50030 
.50079 
.50128 


.34614 
•34636 
•34658 
•34680 
.34702 


.35944 
•35967 
•35989 
.36011 
.36034 


.56114 
.56169 
.56223 
.56278 
.56332 


.37294 
•37317 
.37340 
.37362 
.37385 


.59475 
.59533 
.59590 
.59648 
.59706 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


.33412 
•33434 
•33455 
.33477 
.33499 


.50177 
.50226 
.50275 
.50324 
•50373 


.34724 
.34746 
•34768 
.34790 
.34812 


.53196 
.53247 
.53299 
.53351 
.53403 


.36056 
.36078 
.36101 
•36123 
.36146 


.56387 
.56442 
.56497 
.56551 
.56606 


.37408 
.37430 
.37453 
.37476 
.37498 


.59764 
.59822 
.59880 
.59938 
.59996 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


.33520 
•33542 
•33564 
.33586 
.33607 


. 50422 
.50471 
.50521 
.50570 
.50619 


.34834 
.34856 
.34878 
.34900 
.34923 


.53455 
.53507 
.53559 
.53611 
.53663 


.36168 
.36190 
.36213 
.36235 
•36258 


.56661 
.56716 
.56771 
.56826 
.56881 


.37521 
.37544 
.37567 
.37589 
•37612 


.60054 
.60112 
.60171 
.60229 
.60287 


30 

21 

22 
23 
24 


25 
26 
27 
28 
29 


.33629 
.33651 
.33673 
.33694 
.33716 


.50669 
.50718 
.50767 
.50817 
.50866 


.34945 
.34967 
.34989 
.35011 
.35033 


.53715 
.53768 
.53820 
.53872 
.53924 


.36280 
.36302 
•36325 
.36347 
•36370 


.56937 
.56992 
.57047 
.57103 
.57158 

.57213 
.57269 
.57324 
.57380 
.57436 


•37635 
•37658 
.37680 
.37703 
.37726 


.60346 
.60404 
.60463 
.60521 
•60580 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.33738 
.33760 
.33782 
.33803 
•33825 


.50916 
.50966 
.51015 
.51065 
.51115 


.35055 
.35077 
.35099 
.35122 
.35144 


.53977 
.54029 
.54082 
.54134 
.54187 


•36392 
.36415 
.36437 
•36460 
.38482 


.37749 
.37771 
.37794 
.37817 
.37840 


.60639 
•60698 
•60756 
•60815 
.60874 


30 

31 
32 
33 

34 


35 
36 
37 
38 
39 


•33847 
•33869 
•33891 
•33912 
•33934 


.51165 
.51215 
.51265 
.51314 
.51364 


.35166 
.35188 
.35210 
.35232 
.35254 


. 54240 
.54292 
. 54345 
.54398 
.54451 


.36504 
•36527 
•36549 
•36572 
.36594 

•36617 
•36639 
•36662 
•36684 
.36707 


.57491 
.57547 
.57603 
.57659 
•57715 

.57771 
.57827 
.57883 
.57939 
.57995 


•37862 
.37885 
•37908 
•37931 
.37954 

.37976 
.37999 
.38022 
•38045 
.38068 


.60933 
.60992 
.61051 
.61111 
•61170 

.61229 
.61288 
.61348 
.61407 
.61467 


35 
36 
37 
38 
-39 


40 

41 
42 
43 
44 


.33956 
•33978 
•34000 
•34022 
.34044 


.51415 
.51465 
.51515 
.51565 
.51615 


.35277 
•35299 
.35321 
.35343 
.35365 


.54504 
.54557 
.54610 
.54663 
.54716 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•34065 
.34087 
•34109 
.34131 
•34153 


.51665 
.51716 
.51766 
.51817 
.51867 


.35388 
.35410 
.35432 
•35454 
•35476 


.54769 
.54822 
.54876 
o 54929 
.54982 


•36729 
•36752 
•36775 
•36797 
•36820 


.58051 
.58108 
.58164 
.58221 
•58277 


•38091 
•38113 
•38136 
•38159 
.38182 


.61526 
•61586 
•61646 
.61705 
•61765 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•34175 
•34197 
•34219 
•34241 
•34262 


.51918 
.51968 
.52019 
.52069 
.52120 


•35499 
•35521 
•35543 
.35565 
.35588 


.55036 
.55089 
.55143 
.55196 
.55250 


•36842 
•36865 
•36887 
•36910 
•36932 


.58333 
.58390 
. 58447 
.58503 
•58560 


•38205 
•38228 
•38251 
.38274 
•38296 


.61825 
.61885 
.61945 
•62005 
.62065 
.62125 
.62185 
.62246 
.62306 
•62366 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 


•34284 
.34306 
•34328 
.34350 

•34372 


.52171 
.52222 
.52273 
.52323 
.52374 


•35610 
•35632 
•35654 
•35677 
•35699 


.55303 
.55357 
.55411 
.55465 
.55518 


.36955 
•36978 
•37000 
.37023 
•37045 


.58617 
.58674 
.58731 
.58788 
.58845 


•38319 
.38342 
•38365 
.38388 
.38411 


55 
56 
57 
58 
59 


60 


•34394 


.52425 


•35721 


.55572 


•37068 


.58902 


.38434 


.62427 


60 



730 



tABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTa 
^ 53° 53° 54° 55"* 



« 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


f 




1 

2 
3 

4 


.38434 
.38457 
.38480 
.38503 
•38526 

.38549 
.38571 
.38594 
.38617 
.38640 


.62427 
. 62487 
.62548 
.62609 
-62669 


.39819 
.39842 
.39865 
.39888 
.39911 


.66164 
.66228 
.66292 
.66357 
.66421 
.66486 
.66550 
.66615 
.66679 
.66744 


.41221 
.41245 
.41269 
.41292 
•41316 


.70130 
.70198 
.70267 
.70335 
. 70403 


.42642 
.42666 
.42690 
•42714 
•42738 


. 74345 
.74417 
•74490 
.74562 
•74635 




1 
2 
3 

4 


5 
6 
7 
8 
9 


.62730 
.62791 
.62852 
.62913 
.62974 


.39935 
.39958 
.39981 
.40005 
.40028 


•41339 
•41363 
.41386 
.41410 
.41433 


. 70472 
.70540 
.70609 
.70677 
.70746 


.42762 
.42785 
.42809 
.42833 
•42857 


. 74708 
. 74781 
.74854 
.74927 
•75000 


5 
6 
7 
8 
9 


10 

11 
12 
13 

14 


.38663 
.38686 
.38709 
.38732 
.38755 


.63035 
.63096 
.63157 
.63218 
.63279 


.40051 
.40074 
.40098 
.40121 
.40144 


.66809 
.66873 
.66938 
.67003 
.67068 


.41457 
•41481 
•41504 
•41528 
.41551 


70815 
.70884 
.70953 
.71022 
•71091 


•42881 
.42905 
.42929 
.42953 
•42976 


.75073 
.75146 
•75219 
•75293 
.75366 


10 

11 
12 
13 
14 


15 

16 

17 

18 

19. 

20 

21 

22 

23 

24 


.38778 
.38801 
.38824 
.38847 
.38870 

.38893 
.38916 
.38939 
.38962 
.38985 


.63341 
.63402 
.63464 
.63525 
.63587 
.63648 
.63710 
.63772 
.63834 
.63895 


.40168 
.40191 
.40214 
.40237 
.40261 

.40284 
.40307 
.40331 
.40354 
.40378 


.67133 
.67199 
.67264 
.67329 
.67394 

.67460 
.67525 
.67591 
.67656 
.67722 


.41575 
.41599 
.41622 
.41646 
.41670 

•41693 
•41717 
.41740 
.41764 
.41788 


•71160 
.71229 
.71298 
.71368 
•71437 
.71506 
.71576 
.71646 
.71715 
•71785 


.43000 
.43024 
.43048 
•43072 
•43096 
•43120 
•43144 
•43168 
•43182 
43216 


•75440 
•75513 
•75587 
.75661 
.,•75734 

•75808 
.75882 
.75956 
.76031 
•76105 


15 
16 
17 
18 
-19 
20 
21 
22 
23 
24 


25 
26 
27 
28 
29 


.39009 
.39032 
•39055 
.39078 
.39101 


.63957 
.64019 
.64081 
.64144 
.64206 


.40401 
.40424 
.40448 
.40471 
.40494 


.67788 
.67853 
.67919 
.67985 
•68051 


•41811 
.41835 
.41859 
.41882 
.41906 


.71855 
.71925 
.71995 
.72065 
•72135 


•43240 
•43264 
•43287 
•43311 
•43335 


.76179 
.76253 
.76328 
.76402 
.76477 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.39124 
.39147 
.39170 
.39193 
.39216 


.64268 
.64330 
.64393 
.64455 
.64518 


.40518 
.40541 
.40565 
.40588 
.40611 


.68117 
.68183 
.68250 
.68316 
.68382 


.41930 
.41953 
•41977 
.42001 
.42024 


.72205 
.72275 
.72346 
.72416 
•72487 


.43359 
.43383 
•43407 
•43431 
•43455 


.76552 
.76626 
.76701 
.76776 
.76851 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39. 


.39239 
.39262 
.39286 
.39309 
.39332 
.39355 
.39378 
.39401 
.39424 
.39447 

.39471 
.39494 
.39517 
.39540 
.39563 


.64580 
.64643 
.64705 
.64768 
.64831 

.64894 
.64957 
.65020 
.65083 
.65146 


.40635 
•40658 
.40682 
.40705 
•40728 

.40752 
.40775 
.40799 
.40822 
.40846 


.68449 
.68515 
.68582 
.68648 
.68715 

.68782 
.68848 
.68915 
.68982 
•69049 


.42048 
.42072 
.42096 
•42119 
.42143 


•72557 
•72628 
.72698 
.72769 
. 72840 


.43479 
•43503 
.43527 
•43551 
.43575 


.76926 
•77001 
•77077 
•77152 
•77227 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•42167 
.42191 
.42214 
.42238 
.42262 


•72911 
.72982 
.73053 
.73124 
.73195 


•43599 
•43623 
•43647 
•43671 
•43695 


•77303 
.77378 
.77454 
.77530 
•77606 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


.65209 
.65272 
.65336 
.65399 
.65462 


.40869 
.40893 
.40916 
•40939 
.40963 


.69116 
.69183 
.69250 
.69318 
.69385 

.69452 
.69520 
.69587 
.69655 
.69723 


•42285 
.42309 
.42333 
.42357 
.42381 


.73267 
.73338 
.73409 
.73481 
.73552 


.43720 
.43744 
.43768 
.43792 
.43816 


.77681 
•77757 
•77833 
•77910 
•77986 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


.39586 
.39610 
.39633 
.39656 
.39679 


.65526 
.65589 
.65653 
.65717 
.65780 


.40986 
•41010 
.41033 
.41057 
.41080 


.42404 
.42428 
.42452 
.42476 
.42499 


.73624 
.73696 
.73768 
.73840 
.73911 


•43840 
.43864 
.43888 
.43912 
.43936 


•78062 
•78138 
•78215 
•78291 
.78368 


50 

51 
52 
53 
54 


55 
56 
57 
58 
.59- 


.39702 
.39726 
.39749 
.39772 
.39795 


.65844 
.65908 
.65972 
.66036 
.66100 


.41104 
.41127 
.41151 
.41174 
.41198 


.69790 
.69858 
.69926 
.69994 
.70062 


.42523 
.42547 
.42571 
.42595 
•42619 


.73983 
.74056 
.74128 
•74200 
.74272 


.43960 
•43984 
•44008 
•44032 
•44057 


.78445 
.78521 
•78598 
•78675 
•78752 


55 
56 
57 
58 
59 


60 


.39819 


.66164 


.41221 


.70130 


.42642 


.74345 


.44081 


.78829 


60 



731 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
56° 57*" 58° 59° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


» 




1 

2 
3 
4 


.44081 
.44105 
.44129 
.44153 
.44177 




78829 
78906 
78984 
79061 
79138 


.45536 
.45560 
.45585 
.45609 
.45634 




.83608 
.83690 
.83773 
.83855 
.83938 


.47008 
•47033 
•47057 
•47082 
.47107 


.88708 
.88796 
.88884 
.88972 
•89060 


.48496 
•48521 
•48546 
•48571 
•48596 


.94160 
.94254 
.94349 
.94443 
.94537 




1 
2 
3 
4 


5 
6 
7 
8 
9 


.44201 
.44225 
.44250 
.44274 
.44298 




79216 
79293 
79371 
79449 
79527 


•45658 
•45683 
.45707 
.45731 
.45756 




.84020 
.84103 
84186 
84269 
84352 


.47131 
.47156 
•47181 
.47206 
.47230 


.89148 
.89237 
.89325 
.89414 
•89503 


•48621 
•48646 
•48671 
•48696 
.48721 


.94632 
.94726 
.94821 
.94916 
.95011 


5 
6 
7 
8 
9 


10 

11 
12 
13 
14 


.44322 
.44346 
.44370 
.44395 
.44419 
.44443 
•44467 
.44491 
.44516 
.44540 




79604 
79682 
79761 
79839 
79917 


.45780 
•45805 
•45829 
•45854 
.45878 




84435 
84518 
84601 
84685 
84768 


.47255 
.47280 
.47304 
.47329 
.47354 


.89591 
.89680 
.89769 
.89858 
.89948 


•48746 
•48771 
.48796 
.48821 
.48846 


.95106 
.95201 
.95296 
.95392 
.95487 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 




79995 
80074 
80152 
80231 
80309 


.45903 
.45927 
.45951 
.45976 
•46000 




84852 
84935 
85019 
85103 
85187 


.47379 
.47403 
.47428 
.47453 
.47478 


.90037 
.90126 
.90216 
.90305 
.90395 


.48871 
.48896 
•48921 
•48946 
•48971 


.95583 
.95678 
.95774 
.95870 
.95966 


15 
16 
17 
18 
19 


20 

21 
22 
23 
24 


.44564 
.44588 
.44612 
•44637 
.44661 




80388 
80467 
80546 
80625 
80704 


•46025 
•46049 
•46074 
•46098 
.46123 




85271 
85355 
85439 
85523 
85608 


.47502 
.47527 
.47552 
.47577 
.47601 


.90485 
.90575 
.90665 
.90755 
.90845 


.48996 
.49021 
.49046 
.49071 
.49096 


.96062 
.96158 
.96255 
.96351 
.96448 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.44685 
.44709 
.44734 
•44758 
•44782 




80783 
80862 
80942 
81021 
81101 


.46147 
.46172 
•46196 
.46221 
.46246 




85692 
85777 
85861 
85946 
86031 


.47626 
.47651 
•47676 
•47701 
•47725 


.90935 
.91026 
.91116 
.91207 
.91297 


.49121 
.49146 
.49171 
•49196 
•49221 


.96544 
.96641 
.96738 
.96835 
.96932 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


.44806 
.44831 
•44855 
.44879 
•44903 




81180 
81260 
81340 
81419 
81499 


.46270 
.46295 
.46319 
.46344 
.46368 




86116 
86201 
86286 
86371 
86457 


•47750 
•47775 
•47800 
.47825 
.47849 


.91388 
.91479 
.91570 
.91661 
.91752 


•49246 
•49271 
.49296 
.49321 
.49346 


.97029 
.97127 
.97224 
.97322 
-.97420 


30 

3] 

32 
33 
34 


35 
36 
37 
38 
39 


•44928 
•44952 
•44976 
•45001 
•45025 




81579 
81659 
81740 
81820 
81900 


.46393 
.46417 
.46442 
.46466 
.46491 




86542 
86627 
86713 
86799 
86885 


•47874 
•47899 
.47924 
.47949 
.47974 


.91844 
.91935 
.92027 
.92118 
.92210 


.49372 
•49397 
•49422 
•49447 
•49472 


.97517 
.97615 
.97713 
.97811 
.97910 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•45049 
•45073 
•45098 
•45122 
•45146 




81981 
82061 
82142 
82222 
82303 


.46516 
.46540 
.46565 
.46589 
.46614 




86970 
87056 
87142 
87229 
87315 


.47998 
.48023 
.48048 
.48073 
.48098 


.92302 
.92394 
.92486 
•92578 
.92670 


•49497 
.49522 
.49547 
.49572 
.49597 


.98008 
.98107 
.98205 
.98304 
.98403 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•45171 
•45195 
•45219 
•45244 
■45268 




82384 
82465 
82546 
82627 
82709 


.46639 
.46663 
•46688 
.46712 
.46737 




87401 
87488 
87574 
87661 
87748 


.48123 
.48148 
.48172 
.48197 
•48222 


.92762 
.92855 
.92947 
.93040 
.93133 


.49623 
.49648 
.49673 
•49698 
•49723 


.98502 
.98601 
.98700 
.98799 
.98899 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•45292 
•45317 
.45341 
•45365 
•45390 




82790 
82871 
82953 
83034 
83116 


.46762 
.46786 
.46811 
.46836 
•46860 




87834 
87921 
88008 
88095 
88183 


•48247 
.48272 
.48297 
•48322 
•48347 


.93226 
.93319 
.93412 
.93505 
.93598 


•49748 
.49773 
•49799 
•49824 
•49849 


.98998 
.99098 
.99198 
.99298 
.99398 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59 
60 


•45414 
■45439 
■45463 
■45487 
■45512 
•45536 




83198 
83280 
83362 
83444 
83526 
83608 


•46885 
•46909 
•46934 
.46959 
•46983 
.47008 




88270 
88357 
88445 
88532 
88620 
88708 


•48372 
•48396 
•48421 
.48446 
•48471 
.48496 


.93692 
.93785 
.93879 
.93973 
.94066 

.94160 


•49874 
•49899 
.49924 
•49950 
■49975 
.50000 


.99498 
.99598 
.99698 
.99799 
•99899 
1.00000 


55 
56 
57 
58 
59 
60 



732 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
iV 60° 61° 63° 63° 



/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


t 





.50000 


1^00000 


.51519 


1.06267 


•53053 


1.13005 


.54601 


1.20269 





1 


.50025 


1.00101 


•51544 


1.06375 


•53079 


1.13122 


.54627 


1.20395 


1 


2 


.50050 


1.00202 


•51570 


1.06483 


•53104 


1.13239 


.54653 


1.20521 


2 


3 


•50076 


1.00303 


•51595 


1.06592 


•53130 


1.13356 


.54679 


1.2G647 


3 


, 4 


.50101 


1.00404 


•51621 


1.06701 
1.06809 


.53156 


1.13473 


•54705 


1.20773 


4 


5 


.50126 


1.00505 


•51646 


•53181 


1.13590 


.54731 


1.2C900 


5 


6 


.50151 


1.00607 


.51672 


1.06918 


.53207 


1.13707 


.54757 


1-21C26 


6 


7 


.50176 


1.00708 


.51697 


1.07027 


.53233 


1.13825 


. 54782 


1.21153 


7 


8 


.50202 


1.00810 


.51723 


1.07137 


.53258 


1.13942 


. 54808 


1.21280 


8 


9 


•50227 


1.00912 


.51748 


1.07246 


•53284 


1.14060 


•54834 


1.214C7 


9 


10 


.50252 


1.01014 


.51774 


1.07356 


•53310 


1.14178 


.54860 


1.21535 


10 


11 


.50277 


1.01116 


.51799 


1.07465 


.53336 


1.14296 


.54886 


1.21662 


11 


12 


.50303 


1.01218 


.51825 


1.07575 


•53361 


1.14414 


.54912 


1.21790 


12 


13 


.50328 


1.01320 


.51850 


1.07685 


•53387 


1.14533 


.54938 


1.21918 


13 


14 


.50353 


1.01422 


.51876 


1.07795 
1.07905 


.53413 


1.14651 


.54964 


1-22045 


14 


15 


.50378 


1.01525 


.51901 


.53439 


1.14770 


. 54990 


1-22174 


15 


16 


• 50404 


1.01628 


.51927 


1.08015 


. 53464 


1.14889 


.55016 


1-22302 


16 


17 


• 50429 


1.01730 


.51952 


1.08126 


.53490 


1.15008 


. 55042 


1-22430 


17 


18 


.50454 


1.01833 


.51978 


1.08236 


.53516 


1.15127 


•55068 


1-22559 


18 


19 


.50479 


1.01936 


.52003 


1.08347 


•53542 


1.15246 


.55094 


1.22688 


19 


20 


.50505 


1.02039 


.52029 


1.08458 


.53567 


1.15366 


•55120 


1^22817 


20 


21 


.50530 


1.02143 


.52054 


1.08569 


.53593 


1.15485 


•55146 


1.22946 


21 


22 


.50555 


1.02246 


.52080 


1.08680 


.53619 


1.15605 


•55172 


1.23075 


22 


23 


•50581 


1.02349 


.52105 


1.08791 


.53645 


1.15725 


•55198 


1.23205 


23 


24 


•50606 


1.02453 


.52131 


1.08903 


.53670 


1.15845 


.55224 


1.23334 


24 


25 


•50631 


1.02557 


.52156 


1.09014 


.53696 


1.15965 


.55250 


1.23464 


25 


26 


.50656 


1.02661 


.52182 


1.09126 


.53722 


1.16085 


.55276 


1.23594 


26 


27 


.50682 


1.02765 


.52207 


1.09238 


.53748 


1.16206 


.55302 


1.23724 


27 


28 


.50707 


1.02869 


.52233 


1.09350 


.53774 


1.16326 


.55328 


1.23855 


28 


29 


.50732 


1.02973 


.52259 


1.09462 


.53799 


1.16447 


■55354 


1^23985 


29 


30 


.50758 


1.03077 


.52284 


1.09574 


•53825 


1.16568 


.55380 


1-24116 


30 


81 


.50783 


1.03182 


.52310 


1.09686 


.53851 


1.16689 


.55406 


1.24247 


31 


32 


.50808 


1.03286 


.52335 


1.09799 


.53877 


1.16810 


.55432 


1-24378 


32 


33 


.50834 


1.03391 


.52361 


1.09911 


.53903 


1.16932 


.55458 


1.24509 


33 


34 


•50859 


1.03496 


.52386 


1.10024 


•53928 


1.17053 


.55484 


1^24640 


34 


35 


.50884 


1.03601 


•52412 


1.10137 


.53954 


1-17175 


.55510 


1.24772 


35 


36 


.50910 


1.03706 


•52438 


1.10250 


.53980 


1.17297 


•55536 


1.24903 


36 


37 


.50935 


1.03811 


•52463 


1.10363 


. 54006 


1.17419 


•55563 


1.25035 


37 


38 


.50960 


1.03916 


•52489 


1.10477 


.54032 


1.17541 


•55589 


1.25167 


38 


39 


.50986 


1.04022 


.52514 


1.10590 


•54058 


1.17663 


.5f6]5 


] .25300 


39 


40 


.51011 


1.04128 


.52540 


1.10704 


•54083 


1.17786 


•55641 


1.25432 


40 


41 


.51036 


1.04233 


.52566 


1.10817 


.54109 


1.17909 


•55667 


1.25565 


41 


42 


.51062 


1.04339 


.52591 


1.10931 


.54135 


1.18031 


•55693 


1.25697 


42 


43 


.51087 


1.04445 


.52617 


1.11045 


.54161 


1.18154 


.55719 


1.25830 


43 


44 


.51113 


1.04551 


• 52642 


1.11159 


.54187 


1.18277 


.55745 


1.25963 


44 


45 


.51138 


1.04658 


•52668 


1.11274 


.54213 


1.18401 


•55771 


1.26097 


45 


46 


.51163 


1.04764 


•52694 


1.11388 


.54238 


1-18524 


•55797 


1.26230 


46. 


47 


.51189 


1.04870 


•52719 


1.11503 


. 54264 


1.18648 


.55823 


1.26364 


47 


48 


.51214 


1.04977 


•52745 


1.11617 


. 54290 


1.18772 


.55849 


1.26498 


48 


49 


.51239 


1.05084 


•52771 


1.11732 


.54316 


1.18895 


.55876 


1.26632 


49 


50 


.51265 


1.05191 


•52796 


1.11847 


. 54342 


1.19019 


.55902 


1.26766 


50 


51 


.51290 


1.05298 


•52822 


1.11963 


.54368 


1.19144 


.55928 


1.26900 


51 


52 


.51316 


1.05405 


•52848 


1.12078 


.54394 


1.19268 


.55954 


1-27035 


52 


53 


.51341 


1.05512 


•52873 


1.12193 


54420 


1.19393 


.55980 


1-27169 


53 


54 


.51366 


1.05619 


•52899 


1.12309 


54446 


1.19517 


.56006 


1-27304 


54 


55 


•51392 


1.05727 


.52924 


1.12425 


54471 


1.19642 


.56032 


1-27439 


55 


56 


•51417 


1.05835 


.52950 


1.12540 


54497 


1.19767 


.56058 


L- 27574 


56 


57 


•51443 


1.05942 


.52976 


1.12657 


54523 


1.19892 


.56084 


L.27710 


57 


58 


.51468 


1.06050 


.53001 


1.12773 


54549 


1.20018 


-56111 


L- 27845 


58 


59 


•51494 


1.06158 


53027 


1.12889 


54575 


1-20143 


•56137 


L .27981 


St 


60 


.51519 


1.06267 


•53053 


1-13005 


54601 


1.20269 


.56163 


L-28117 


60 



733 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
64° 65° 66° 67° 



f 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


t 




1 

2 
3 
4 


.56163 
.56189 
.56215 
.56241 
.56267 


1.28117 
1.28253 
1.28390 
1.28526 
1.28663 


.57738 
.57765 
.57791 
.57817 
.57844 


1.36620 
1.36768 
1.36916 
1.37064 
1-37212 


.59326 
.59353 
.59379 
.59406 
. 59433 


1.45859 
1.46020 
1.46181 
1.46342 
1.46504 


.60927 
.60954 
.60980 
.61007 
.61034 




55930 
56106 
56282 
56458 
56634 


O 

1 

2 
3 

4 


5 
6 
7 
8 
9 


.56294 
.56320 
.56346 
.56372 
.56398 


1.28800 
1.28937 
1-29074 
1.29211 
1.29349 


.57870 
.57896 
.57923 
.57949 
.57976 


1.37361 
1.37509 
1.37658 
1.37808 
1.37957 


.59459 
.59486 
.59512 
.59539 
.59566 


1.46665 
1.46827 
1.46989 
1.47152 
1.47314 


.61061 
.61088 
.61114 
.61141 
.61168 




56811 
56988 
57165 
57342 
57520 


3 
6 
7 
8 
9 


10 

11 
12 
13 

14 


.56425 
.56451 
.56477 
.56503 
.56529 


1.29487 
1.29625 
1.29763 
1.29901 
1.30040 


.58002 
.58028 
.58055 
.58081 
.58108 


1.38107 
1.38256 
1.38406 
1.38556 
1.38707 


.59592 
.59619 
.59645 
.59672 
.59699 


1.47477 
1.47640 
1.47804 
1.47967 
1.48131 


•61195 
.61222 
.61248 
.61275 
.61302 




57698 
57876 
58054 
58233 
58412 


10 

11 

12 

13 
14 


15 
16 
17 
18 
19 


.56555 
.56582 
.56608 
.56634 
.56660 


1.30179 
1.30318 
1.30457 
1.30596 
1.30735 


.58134 
.58160 
•58187 
•58213 
•58240 


1.38857 
1.39008 
1.39159 
1.39311 
1.39462 


.59725 
•59752 
.59779 
•59805 
-59832 


1.48295 
1.48459 
1.48624 
1.48789 
1.48954 


•61329 
.61356 
.61383 
.61409 
•61436 




58591 
58771 
58950 
59130 
59311 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


.56687 
.56713 
.56739 
.56765 
.56791 


1.30875 
1.31015 
1.31155 
1.31295 
1.31438 


•58266 
•58293 
58319 
58345 
58372 


1.39614 
1.39766 
1.39918 
1.40070 
1.40222 


-59859 
59885 
-59912 
-59938 
-59965 


1.49119 
1.49284 
1.49450 
1.49616 
1.49782 


.61463 
.61490 
.61517 
.61544 
.61570 




59491 
59672 
59853 
60035 
60217 


30 

21 
22 
23 
24 


25 
26 
27 
28 
29 


.56818 
•56844 
.56870 
.56896 
.56923 


1.31576 
1.31717 
1.31858 
1.31999 
1.32140 


58398 
58425 
58451 
58478 
58504 


1.40375 
1.40528 
1.40681 
1.40835 
1.40988 


59992 
-60018 
-60045 
.60072 
.60098 


1.49948 
1.50115 
1.50282 
1.50449 
1.50617 


.61597 
.61624 
.61651 
.61678 
.61705 




60399 
60581 
60763 
60946 
61129 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•56949 
.56975 
.57001 
.57028 
.57054 


1.32282 
1.32424 
1.32566 
1.32708 
1.32850 


58531 
58557 
58584 
58610 
58637 


1.41142 
1.41296 
1.41450 
1.41605 
1.41760 


60125 
60152 
60178 
60205 
60232 


1.50784 
1.50952 
1.51120 
1.51289 
1.51457 


.61732 
.61759 
.61785 
.61812 
.61839 




61313 
61496 
61680 
61864 
62049 


30 

31 
3^ 
33 
34 


35 
36 
37 
38 
39,, 


• 57080 
.57106 
.57133 
.57159 
.57185 


1.32993 
1.33135 
1.33278 
1.33422 
1.33565 


58663 
58690 
58716 
58743 
58769 


1.41914 
1.42070 
1.42225 
1.42380 


.60259 
.60285 
.60312 
.60339 
.60365 


1.51626 
1.51795 
1.51965 
1.52134 
1.52304 


.61866 
.61893 
.61920 
.61947 
.61974 




62234 
62419 
62604 
62790 
62976 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


.57212 
.57238 
•57264 
.57291 
.57317 


1-33708 
1-33852 
1^33996 
1.34140 
1.34284 


-58796 
-58822 
-58849 
-58875 
-58902 


1.42692 
1.42848 
1.43005 
1.43162 
1.43318 


.60392 
.60419 

• 60445 

• 60472 
•60499 


1.52474 
1.52645 
1.52815 
1.52986 
1.53157 


.62001 
.62027 
•62054 
.62081 
.62108 




63162 
63348 
63535 
63722 
63909 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


.57343 
.57369 
•57396 
•57422 
•57448 


1.34429 

1.34573 

1.34718 

1.34863 

1.35009 

1.35154 

1.35300 

1.35446 

1.35592 

1.35138_ 

1.35885 

1.36031 

1.36178 

1.36325 

1-36473 


-58928 
-58955 
-58981 
•59008 
•59034 


1.43476 
1.43633 
1.43790 
1.43948 
1.44106 


.60526 
.60552 
.60579 
.60606 
•60633 


1.53329 
1.53500 
1.53672 
1.53845 
1.54017 


.62135 
.62162 
.62189 
•62216 
• 62243 
•62270 
•62297 
•62324 
•62351 
•62378 




64097 
64285 
64473 
64662 
64851 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•57475 
.57501 
.57527 
.57554 
•57580 


•59061 
•59087 
•59114 
•59140 
.59167 


1.44264 
1.44423 
1.44582 
1.44741 
1.44900 


.60659 
•60686 
.60713 
.60740 
.60766 
.60793 
.60820 
.60847 
.60873 
.60900 


1.54190 
1.54363 
1.54536 
1.54709 
1.54883 




65040 
65229 
65419 
65609 
65799 


50 

51 
52 
53 
54 


55 
56 
57 
58 
59,. 


.57606 
.57633 
.57659 
.57685 
•57712 


•59194 
•59220 
•59247 
•59273 
•59300 


1.45059 
1.45219 
1.45378 
1.45539 
1.45699 


1.55057 
1-55231 
1-55405 
1.55580 
1-55755 


•62405 
.62431 
.62458 
.62485 
-62512 




65989 
66180 
66371 
66563 
66755 


55 
56 
57 
58 
59 


60 


.57738 


1.36620 


•59326 


1.45859 


.60927 


1.55930 


•62539 


1 


66947 


60 



734 



\ 

r^BLE X.— NATURAL VERSED SINES AND EXTERNAL 
/\ 68° 69° 70° 7r 


SECANTS. 


f 

"o" 

1 
2 
3 
4 
5 
6 
7 
8 
9 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


^ Ex. sec. 


Vers. 


Ex. sec. 


1 


.62539 
.62566 
•62593 
•62620 
.62647 


1^ 66947 
1.67139 
1^67332 
1^67525 
1^67718 


•64163 
•64190 
•64218 
•64245 
.64272 




79043 
79254 
79466 
79679 
79891 


•65798 
•65825 
.65853 
•65880 
•65907 




92380 
92614 
92849 
93083 
93318 


•67443 2 
• 67471 12 
•67498 2 
•67526 2 
.67553 2 


07155 
07415 
07675 
07936 
08197 




1 

2 
3 
4 


.62674 
.62701 
.62728 
.62755 
.62782 


1.67911 
1.68105 
1.68299 
1.68494 
1.68689 


.64299 
.64326 
•64353 
•64381 
• 64408 




80104 
80318 
80531 
80746 
80960 


.65935 
.65962 
.65989 
.66017 
• 66044 


1 


93554 
93790 
94026 
94263 
94500 


.67581 \2 
.67608 \2 
.67636 2 
.67663 12 
•67691 12 


08459 
08721 
08983 
09246 
09510 


5 
6 
7 
8 

9 


10 

11 
12 
13 

14 


•62809 
•62836 
.62863 
.62890 
.62917 


1.68884 
1.69079 
1.69275 
1.69471 
1.69667 


•64435 

• 64462 

• 64489 
.64517 
. 64544 




81175 
81390 
81605 
81821 
82037 


•66071 
•66099 
•66126 
•66154 
•66181 




94737 
94975 
95213 
95452 
95691 


•67718 
•67746 
.67773 
.67801 
.67829 


2 
2 
2 
2 
2 


09774 
10038 
10303 
10568 
10834 


10 

11 

12 
13 
14 


15 
16 
17 
18 
19 


• 62944 
•62971 
•62998 
•63025 
•63052 


1.69864 
1.70061 
1.70258 
1.70455 
1.70653 


.64571 
.64598 
.64625 
.64653 
.64680 




82254 
82471 
82688 
82906 
83124 


.66208 
.66236 
.66263 
•66290 
•66318 




95931 
96171 
96411 
96652 
96893 


.67856 
.67884 
.67911 
.67939 
•67966 


2 
2 
2 
2 
2 


11101 
11367 
11635 
11903 
12171 


15 
16 
17 
18 
19 


30 

21 
,22 
23 
,24 

(25 
'26 
27 
(28 
29 
130 
(31 
32 
133 
34 

i35 
36 
37 
138 
;39 
10 
;41 
42 
43 
144 

45 
'46 

47 
"48 
,49 
^0 
,^1 

53 
54 


•63079 
•63106 
•63133 
•63161 
•63188 


1.70851 
1.71050 
1.71249 
1.71448 
1.71647 


. 64707 
. 64734 
.64761 
.64789 
•64816 




83342 
83561 
83780 
83999 
84219 


•66345 
•66373 
• 66400 
•66427 
•66455 


1 


97135 
97377 
97619 
97862 
98106 


•67994 
•68021 
•68049 
•68077 
•68104 


2 
2 
2 
2 
2 


12440 
12709 
12979 
13249 
13520 


30 

21 
22 
23 
24 


•63215 
•63242 
•63289 
•63296 
.63323 


1.71847 
1.72047 
1 . 72247 
1.72448 
1.72649 


• 64843 
. 64870 
•64898 
•64925 
•64952 




84439 
84659 
84880 
85102 
85323 


• 66482 
•66510 
•66537 
•66564 
•66592 




98349 
98594 
98838 
99083 
99329 


•68132 
.68159 
•68187 
•68214 
•68242 


2 
2 
2 
2 
2 


13791 
14063 
14335 
14608 
14881 


25 
26 
27 
28 
29 


•63350 
.63377 
. 63404 
.63431 
.63458 


1.72850 
1.73052 
1.73254 
1.73456 
1.73659 


•64979 
.65007 
•65034 
•65061 
•65088 




85545 
85767 
85900 
86213 
86437 


•66619 
•66647 
• 66674 
•66702 
•66729 


2 
2 
2 


99574 
99821 
00067 
00315 
00562 


•68270 
•68297 
-68325 
-68352 
-68380 


2 
2 
2 
2 
2 


15155 
15429 
15704 
15979 
16255 


30 

aa 

3!2 
33 
34 


•63485 
•63512 
•63539 
•63566 
•63594 


1.73862 
1.74065 
1.74269 
1 . 74473 
1.74677 


•65116 
-65143 
•65170 
•65197 
•65225 


1 


86661 
86885 
87109 
87334 
87560 


•66756 
•66784 
•66811 
-66839 
•66866 


2 
2 
2 
2 
2 


00810 
01059 
01308 
01557 
01807 


- 68408 
68435 
-68463 
-68490 
•68518 


2 
2 
2 
2 

L 


16531 
16808 
17085 
17363 
17641 


35 
36 
37 
38 
39 


•63621 
•63648 
•63675 
•63702 
•63729 


1.74881 
1.75086 
1.75292 
1.75497 
1.75703 


•65252 
•65279 
.65306 
.65334 
.65361 




87785 
88011 
88238 
88465 
88692 


•66894 
.66921 
-66949 
•66976 
•67003 


2 
2 
2 
2 
2 


02057 
02308 
02559 
02810 
03062 


•68546 
.68573 
.68801 
•68628 
-68656 


2 
2 
2 
2 
2 


17920 
18199 
18479 
18759 
19040 


40 

41 
42 
43 
44 


•63756 
•63783 
•63810 
•63838 
•63865 
•63892 
•63919 
•63946 
•63973 
• 64000 


1.75909 
1.76116 
1.76323 
1.76530 
1.76737 


.65388 
.65416 
.65443 
.65470 
•65497 




88920 
89148 
89376 
89605 
89834 


•67031 
•67058 
.67086 
•67113 
•67141 


2 
2 
2 
2 
2 


03315 
03568 
03821 
04075 
04329 


•68684 
•68711 
•68739 
.68767 
•68794 


2 
2 
2 
2 
2 


19322 
19604 
19886 
20169 
20453 


45 
46 
47 
48 
49 


1.76945 
1.77154 
1.77362 
1.77571 
1.77780 


•65525 
•65552 
•65579 
•65607 
•65634 




90063 
90293 
90524 
90754 
90986 


•67168 
•67196 
.67223 
•67251 
•67278 


2 
2 
2 
2 
2 


04584 
04839 
05094 
05350 
05607 


•68822 
• 68849 
•68877 
•68905 
•68932 


2 
2 
2 
2 
2 


20737 
21021 
21306 
21592 
21878 


50 

51 
52 
53 
54 


I55 
156 
;57 
,58 


•64027 
•64055 
•64082 
-64109 
•64136 


1.77990 
1.78200 
1.78410 
1.78621 
1.78832 


•65661 
•65689 
•65716 
-65743 
•65771 




91217 
91449 
91681 
91914 
92147 


.67306 
.67333 
.67361 
.67388 
•67416 


2 
2 
2 
2 
2 


05864 
06121 
06379 
06637 
06896 


.68960 
.68988 
.69015 
. 69043 
•69071 


2 
2 
2 
2 
2 


22165 
22452 
22740 
23028 
23317 


55 
56 
57 
58 
59 


fco 


• 64ij63 


1.79043 


.65798 


1 


92380 


•67443 


2 


07155 


•69098 


2.23607 


60 



735 



TABLE X.— NATURAL VERSED STNES AND EXTERNAL SECANTS. 
73° 73° 74° 75° 


f 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 




1 
2 
3 
4 


.69098 
.69126 
.69154 
.69181 
.69209 


2.23607 
2.23897 
2-24187 
2.24478 
2-24770 


.70763 
.70791 
.70818 
. 70846 
.70874 


2.42030 
2.42356 
2.42683 
2.43010 
2.43337 


-72436 
-72464 
-72492 
-72520 
-72548 


2-62796 
2-63164 
2-63533 
2-63903 
2-64274 


-74118 
- 74146 
-74174 
. 74202 
.74231 


2-86370 
2-86790 
2-87211 
2-87633 
2-88056 




1 
2 
3 
4 

5 
6 
7 
8 

9 


5 
6 
7 
8 

9 


.69237 
.69264 
.69292 
.69320 
.69347 


2.25062 
2.25355 
2.25648 
2.25942 
2.26237 


.70902 
.70930 
.70958 
.70985 
.71013 


2.43666 
2-43995 
2.44324 
2.44655 
2.44986 


-72576 
-72604 
-72632 
.72660 
.72688 


2-64645 
2-65018 
2-65391 
2-65765 
2-66140 


. 74259 
.74287 
.74315 
. 74343 
.74371 


2-88479 
2-88904 
2-89330 
2.89756 
2-90184 


10 

11 
12 
13 
14 


•69375 
. 69403 
•69430 
•69458 
•69486 


2.26531 
2.26827 
2.27123 
2.27420 
2.27717 


.71041 
.71069 
.71097 
.71125 
.71153 


2.45317 
2.45650 
2.45983 
2.46316 
2.46651 


.72716 
•72744 
.72772 
.72800 
.72828 


2.66515 
2-66892 
2.67269 
2.67647 
2.68025 


.74399 
.74427 
. 74455 
. 74484 
.74512 


2-90613 
2-91042 
2-91473 
2.91904 
2-92337 


10 

11 
12 
13 
14 


15 
16 
17 
18 
19 


•69514 

.69541 

69569 

69597 

•69624 


2.28015 
2.28313 
2.28612 
2.28912 
2.29212 


.71180 
.71208 
.71236 
.71264 
.71292 


2.46986 
2.47321 
2.47658 
2.47995 
2.48333 


.72856 
.72884 
.72912 
. 72940 
.72968 


2.68405 
2-68785 
2-69167 
2-69549 
2-69931 


. 74540 
.74568 
.74596 
. 74624 
.74652 


2-92770 
2-93204 
2-93640 
2-94076 
2.94514 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


•69652 
•69680 
•69708 
.69735 
.69763 


2^29512 
2.29814 
2.30115 
2.30418 
2.30721 


.71320 
•71348 
.71375 
.71403 
.71431 


2.48671 
2.49010 
2.49350 
2.49691 
2-50032 


.72996 
.73024 
.73052 
.73080 
.73108 


2-70315 
2-70700 
2.71085 
2^71471 
2^71858 


. 74680 
. 74709 
.74737 
.74765 
.74793 


2.94952 
2.95392 
2-95832 
2-96274 
2-96716 


20 

21 
22 
23 
24 


25 
26 
27 
28 
29 


•69791 
•69818 
•69846 
•69874 
•69902 


2.31024 
2.31328 
2.31633 
2.31939 
2^32244 


.71459 
.71487 
.71515 
.71543 
.71571 


2-50374 
2-50716 
2-51060 
2-51404 
2-51748 


.73136 
.73164 
.73192 
.73220 
-73248 


2.72246 
2.72635 
2.73024 
2.73414 
2.73806 


.74821 
. 74849 
.74878 
. 74906 
. 74934 


2-97160 
2-97604 
2-98050 
2.98497 
2-98944 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•69929 
•69957 
•69985 
•70013 
• 70040 


2-32551 
2-32858 
2.33166 
2.33474 
2.33783 


.71598 
.71626 
•71654 
•71682 
.71710 


2.52094 
2-52440 
2-52787 
2.53134 
2-53482 


-73276 
•73304 
•73332 
.73360 
.73388 


2.74198 
2.74591 
2.74984 
2.75379 
2-75775 


.74962 
- 74990 
-75018 
•75047 
.75075 


2-99393 
2-99843 
3^00293 
3^00745 
3.01198 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


.70068 
.70096 
.70124 
•70151 
•70179 


2-34092 
2.34403 
2-34713 
2-35025 
2^35336 


•71738 
.71766 
•71794 
.71822 
.71850. 


2-53831 
2.54181 
2-54531 
2-54883 
2-55235 


•73416 
• 73444 
.73472 
.73500 
•73529 


2-76171 
2.76568 
2-76966 
2-77365 
2-77765 


.75103 
-75131 
-75159 
-75187 
•75216 


3-01652 
3-02107 
3-02563 
3-03020 
3-03479 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•70207 
•70235 
• 70263 
•70290 
•70318 


2-35649 
2-35962 
2-36276 
2.36590 
2.36905 


.71877 
.71905 
.71933 
•71961 
•71989 


2-55587 
2.55940 
2.56294 
2.56649 
2.57005 


•73557 
•73585 
•73613 
•73641 
•73669 


2-78166 
2-78568 
2-78970 
2-79374 
2-79778 


- 75244 
•75272 
•75300 
-75328 
•75356 


3 •03938 
3 • 04398 
3-04860 
3-05322 
3-05786 


40 

41 

42 
43 
44 
45 
46 
47 
48 
49 


45 
46 
47 
48 
49 


•70346 
•70374 
•70401 
. 70429 
•70457 


2-37221 
2.37537 
2.37854 
2.38171 
2.38489 


•72017 
. 72045 
•72073 
•72101 
.72129 


2.57361 
2.57718 
2.58076 
2-58434 
2.58794 


•73697 
•73725 
•73753 
•73781 
•73809 


2-80183 
2-80589 
2.80996 
2-81404 
2-81813 


-75385 
-75413 
- 75441 
.75469 
.75497 


3-06251 
3-06717 
3-07184 
3-07652 
3-08121 


50 

51 
52 
53 
54 


•70485 
•70513 
•70540 
•70568 
.70596 


2.38808 
2.39128 
2.39448 
2.39768 
2.40089 


.72157 
.72185 
.72213 
•72241 
.72269 


2.59154 
2.59514 
2.59876 
2.60238 
2.60601 


.73837 
•73865 
.73893 
•73921 
•73950 


2^82223 
2-82633 
2-83045 
2-83457 
2^83871 


.75526 
-75554 
-75582 
-75610 
•75639 


3^08591 
3^09063 
3^09535 
3^10009 
3^10484 


50 

51 
52 
53 
54 
55 
56 
57 
58 
59 


55 
56 
57 
58 
59 


.70624 
•70652 
.70679 
.70707 
•70735 


2-40411 
2.40734 
2-41057 
2.41381 
2.41705. 


.72296 
.72324 
.72352 
.72380 
•72408 


2.60965 
2.61330 
2.61695 
2-62061 
2-62428 


•73978 
•74006 
• 74034 
•74062 
•74090 


2-84285 
2-84700 
2-85116 
2.85533 
2^85951 


•75667 
•75695 
•75723 
-75751 
•75780 


3-10960 
3-11437 
3-11915 
3-12394 
3-12875 


60 


•70763 


2.42030 


.72436 


2-62796 


.74118 


2-86370 


.75808 


3-13357 


60 










71 


J6 











^ TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 
76° 77° 78° 79° 





Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


t 


w 


•75808 


3^13357 


•77505 


3^44541 


-79209 


3-80973 


•80919 


4^24084 





•75836 


3 


•13839 


•77533 


3^45102 


•79237 


3-81633 


•80948 


4-24870 


1 


•75864 


3 


•14323 


•77562 


3.45664 


•79266 


3-82294 


•80976 


4-25658 


2 


■ 5 


•75892 


3 


•14809 


•77590 


3^46228 


•79294 


3-82956 


•81005 


4-26448 


3 


•75921 


3 


15295 


•77618 


3-46793 


•79323 


3-83621 


•81033 


4-27241 


4 


•75949 


3 


15782 


•77647 


3-47360 


•79351 ^ 


3-84288 


•81062 


4-28036 


5 


6 


•75977 


3 


16271 


•77675 


3-47928 


•79380 


3-84956 


•81090 


4-28833 


R 


s 


•76005 


3 


16761 


. 77703 


3 48498 


• 79408 


3-85627 


•81119 


4-29634 


7 


•76034 


3 


17252 


•77732 


3^49069 


•79437 


3.86299 


•81148 


4-30436 


8 


f 9 


•76062 


3 


17744 


•77760 


3^49642 


•79465 


3-86973 


-81176 


4.31241 


9 


flO 


•76090 


3 


18238 


•77788 


3 • 50216 


- 79493 


3.87649 


-81205 


4-32049 


10 


11 


•76118 


3 


18733 


•77817 


3^50791 


-79522 


3.88327 


-81233 


4-32859 


11 


h2 


•76147 


3 


19228 


•77845 


3^51368 


.79550 


3.89007 


-81262 


4-33671 


12 


cl3 


•76175 


3 


19725 


•77874 


3^51947 


-79579 


3.89689 


-81290 


4-34486 


13 


[l^ 


•76203 


3 


20224 


•77902 


3^52527 


-79607 


3-90373 


-81319 


4-35304 


14 


:15 


•76231 


3 


20723 


•77930 


3.53109 


.79636 


3.91058 


-81348 


4-36124 


15 


1« 


•76260 


3 


21224 


•77959 


3.53692 


.79664 


3-91746 


-81376 


4-36947 


16 


Si7 


•76288 


3 


21726 


•77987 


3 . 54277 


.79693 


3^92436 


-81405 


4-37772 


17 


i« 


•76316 


3 


22229 


•78015 


3 . 54863 


.79721 


3^93128 


•81433 


4-38600 


18 


^19 


•76344 


3 


22734 


. 78044 


3-55451 
3.56041 


.79750 
•79778 


3-93821 
3-94517 


-81462 
•81491 


4-39430 
4-40263 


19 


■30 


•76373 


3 


23239 


.78072 


30 


.'2i 


•76401 


3 


23746 


•78101 


3.56632 


•79807 


3-95215 


•81519 


4^41099 


2. 


,22 


•76429 


3 


24255 


.78129 


3.57224 


•79835 


3-95914 


•81548 


4^41937 


2^ 


123 


•76458 


3 


24764 


.78157 


3.57819 


•79864 


3-96616 


•81576 


4^42778 


23 


25 


• 76486 


3 


25275 


•78186 


3-58414 


-79892 


3-97320 


•81605 


4.43622 


24 


•76514 


3 


25787 


. 78214 


3-59012 


•79921 


3-98025 


•81633 


4-44468 


25 


'26 


•76542 


3 


26300 


. 78242 


3^59611 


•79949 


3-98733 


•81662 


4-45317 


26 


27 


•76571 


3 


26814 


.78271 


3 • 60211 


•79978 


3-99443 


•81691 


4-46169 


27 


28 


•76599 


3 


27330 


.78299 


3-60813 


•80006 


4-00155 


•81719 


4-47023 


28 


29 
30 


•76627 


3 


27847 


•78328 


3-61417 


•80035 


4-00869 


-81748 


4^47881 


29 


•76655 


3 


28366 


•78356 


3-62023 


•80063 


4-01585 


-81776 


4 •48740 


30 


31 


•76684 


3 


.28885 


•78384 


3^62630 


•80092 


4-02303 


-81805 


4^49603 


31 


32 


•76712 


3 


.29406 


•78413 


3.63238 


.80120 


4-03024 


-81834 


4^50468 


32 


33 


• 76740 


3 


29929 


• 78441 


3.63849 


.80149 


4-03746 


•81862 


4^51337 


33 


34 


•76769 


3 


30452 


• 78470 


3 - 64461 


•80177 


4-04471 


-81891 


4^52208 


34 


35 


•76797 


3 


30977 


• 78498 


3-65074 


•80206 


4-05197 


•81919 


4^53081 


35 


36 


•76825 


3 


31503 


•78526 


3^65690 


•80234 


4-05926 


-81948 


4^53958 


36 


37 


•76854 


3 


32031 


•78555 


3.66307 


.80263 


4-06657 


-81977 


4^54837 


37 


38 


•76882 


3 


32560 


•78583 


3.66925 


•80291 


4-07390 


-82005 


4^55720 


38 


39 


•76910 


3 


33090 


•78612 


3-67545 


-80320 


4.08125 


•82034 


4.56605 


39 


40 


•76938 


3. 


33622 


.78640 


3-68167 


-80348 


4.08863 


•82063 


4^57493 


40 


41 


76967 


3^ 


34154 


•78669 


3.68791 


-80377 


4-09602 


•82091 


4^58383 


41 


42 


76995 


3^ 


34689 


•78697 


3.69417 


-80405 


4.10344 


•82120 


4^59277 


42 


43 


77023 


3^ 


35224 


•78725 


3 . 70044 


-80434 


4.11088 


•82148 


4^60174 


43 


44 
45 


77052 


3. 


35761 


•78754 


3.70673 


-80462 
-80491 


4-11835 
4.12583 


-82177 
-82206 


4-61073 


44 


77080 


S^ 


36299 


.78782 


3-71303 


4-61976 


45 


46 


77108 


3^ 


36839 


.78811 


3.71935 


-80520 


4.13334 


•82234 


4^ 62881 


46 


47 


77137 


3. 


37380 


.78839 


3.72569 


-80548 


4.14087 


•82263 


4^63790 


47 


48 


77165 


3- 


37923 


.78868 


3.73205 


-80577 


4-14842 


-82292 


4-64701 


48 


49 
50 


77193 


3. 


38466 


•78896 
•78924 


3.73843 


-80605 


4^15599 


82320 


4-65616 


49 


77222 


3^ 


39012 


3 . 74482 


-80634 


4.16359 


-82349 


4-66533 


50 


51 


77250 


3^ 


39558 


•78953 


3.75123 


-80662 


4.17121 


.82377 


4-67454 


51 


52 


77278 


3^ 


40106 


•78981 


3.75766 


.80691 


4^17886 


-82406 


4-68377 


52 


53 


77307 


3^ 


40656 


•79010 


3.76411 


•80719 


4^18652 


•82435 


4-69304 


53 


54 
55 


77335 


3^ 


41206 


•79038 
•79067 


3.77057 


-80748 


4-1.9421 


•82463 


4-70234 


54 


77363 


3. 


41759 


3^77705 


80776 


4-20193 


-82492 


4-71166 


55 


56 


77392 


3^ 


42312 


•79095 


3 •78355 


80805 


4-20966 


•82521 


4-72102 


56 


57 


•77420 


3^ 


42867 


•79123 


3-79007 


80833 


4-21742 


•82549 


4-73041 


57 


58 


• 77448 


3. 


43424 


•79152 


3-79661 


80862 


4-22521 


82578 


4^73983 


58 


59 


•77477 


3 


43982 


•79180 


3^80316 


80891 


4-23301 


82607 


4^74929 


59 


60 


•77505 


3 


44541 


.79209 


3-80973 


80919 


4-24084 


82635 


4-75877 


60 



737 



TABLE X.— NATURAL VERSED SINES AND EXTERNAL SECANTS. 

80° 81° 83° 83° 


/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


t 




1 
2 
3 
4 


.82635 
.82664 
.82692 
.82721 
.82750 


4.75877 
4.76829 
4.77784 
4.78742 
4.79703 


-84357 
.84385 

.84414 
.84443 
. 84471 


5.39245 
5-40422 
5-41602 
5-42787 
5-43977 


-86083 
-86112 
-86140 
-86169 
.86198 
-86227 
-86256 
-86284 
.86313 
•86342 


6.18530 
6.20020 
6.21517 
6.23019 
6.24529 


-87813 
.87842 
-87871 
-87900 
.87929 


7.20551 
7.22500 
7.24457 
7.26425 
7-28402 




1 
2 
3 

4 


5 
6 
7 
8 
9 


•82778 
82807 
.82836 
•82864 
•82893 


4.80667 
4.81635 
4-82606 
4-83581 
4-84558 


-84500 
.84529 
-84558 
.84586 

.84615 


5-45171 
5-46369 
5.47572 
5.48779 
5-49991 


6-26044 
6-27566 
6-29095 
6-30630 
6-32171 


-87957 
-87986 
-88015 
-88044 
.88073 


7-30388 
7-32384 
7-34390 
7-36405 
7-38431 


5 

6 
7 
8 
9 


10 

11 
12 
13 

14 


•82922 
•82950 
.82979 
.83008 
•83036 


4.85539 
4.86524 
4.87511 
4.88502 
4.89497 


.84644 
-84673 
-84701 
-84730 
.84759 
-84788 
-84816 
-84845 
.84874 
.84903 

.84931 
.84960 
-84989 
-85018 
-85046 


5-51208 
5-52429 
5-53655 
5-54886 
5.56121 


.86371 
.86400 
•86428 
.86457 
.86486 


6-33719 
6-35274 
6-36835 
6.38403 
6.39978 


.88102 
.88131 
.88160 
.88188 
.88217 


7-40466 
7-42511 
7.44566 
7-46632 
7-48707 


10 

11 

12 
13 
14 


15 
16 
17 
18 
19 


.83065 

.83094 

.83122 

.83151 

•83J8(L 

•83208 

•83237 

•83266 

•83294 

.83323 


4.90495 
4.91496 
4.92501 
4.93509 
4.94521 
4-95536 
4.96555 
4.97577 
4.98603 
4-99633 


5-57361 
5-58606 
5-59855 
5.61110 
5.62369, 


.86515 
.86544 
.86573 
.86601 
•86630 


6-41560 
6-43148 
6-44743 
6.46346 
6.47955 


-88246 
-88275 
.88304 
.88333 
.88362 


7-50793 
7-52889 
7-54996 
7.57113 
7-59241 


15 
16 
17 
18 
19 


30 

21 
22 
23 
24 


5-63633 
5-64902 
5-66176 
5-67454 
5-68738 


•86659 
.86688 
-86717 
-86746 
.86774 


6.49571 
6-51194 
6-52825 
6-54462 
6-56107 


.88391 
.88420 
. 88448 
.88477 
.88506 


7-61379 
7-63528 
7-65688 
7-67859 
7-70041 


30 

21 

22 
23 
24 


25 
26 
27 
28 
29 


.83352 
83380 
.83409 
.83438 
•83467 


5-00666 
5-01703 
5-02743 
5-03787 
5.04834 


-85075 
.85104 
.85133 
•85162 
.85190 


5-70027 
5.71321 
5-72620 
5-73924 
5-75233 


-86803 
-86832 
-86861 
-86890 
.86919 


6-57759 
6-59418 
6-61085 
6-62759 
6-64441 


.88535 
.88564 
-88593 
.88622 
.88651 


7-72234 
7.74438 
7.76653 
7.78880 
7-81118 


25 
26 
27 
28 
29 


30 

31 
32 
33 
34 


•83495 
83524 
•83553 
•83581 
•83610 


5.05886 
5.06941 
5.08000 
5.09062 
5.10129 


.85219 
.85248 
-85277 
-85305 
.85334 


5.76547 
5.77866 
5.79191 
5-80521 
5-81856 


-86947 
.86976 
.87005 
.87034 
.87063 

.87092 
.87120 
.87149 
.87178 
.87207 


6-66130 
6.67826 
6.69530 
6-71242 
6-72962 


.88680 
.88709 
.88737 
-88766 
.88795 


7-83367 
7-85628 
7-87901 
7-90186 
7-92482 


30 

31 
32 
33 
34 


35 
36 
37 
38 
39 


•83639 
•83667 
•83696 
•83725 
•83754 


5.11199 
5-12273 
5-13350 
5-14432 
5.15517 


.85363 
-85392 
.85420 
.85449 
.85478 


5-83196 
5-84542 
5-85893 
5-87250 
5.88612 


6.74689 
6.76424 
6.78167 
6-79918 
6-81677 


.88824 
.88853 
-88882 
-88911 
. 88940 


7-94791 
7-97111 
7-99444 
8-01788 
8-04146 


35 
36 
37 
38 
39 


40 

41 
42 
43 
44 


•83782 
•83811 
•83840 
•83868 
•83897 


5^16607 
5^17700 
5-18797 
5-19898 
5-21004 


.85507 
.85536 
-85564 
-85593 
.85622 


5.89979 
5.91352 
5.92731 
5-94115 
5.95505 


.87236 
•87265 
•87294 
•87322 
.87351 


6-83443 
6-85218 
6-87001 
6-88792 
6-90592 


.88969 
.88998 
.89027 
.89055 
.89084 


8-06515 
8-08897 
8-11292 
8-13699 
8-16120 


40 

41 
42 
43 
44 


45 
46 
47 
48 
49 


•83926 
.83954 
.83983 
•84012 
• 84041 


5-22113 
5-23226 
5-24343 
5-25464 
5-26590 


-85651 
.85680 
.85708 
-85737 
-85766 


5.96900 
5.98301 
5.99708 
6.01120 
6^02538 


.87380 
.87409 
•87438 
•87467 
•87496 


6-92400 
6-94216 
6-96040 
6-97873 
6-99714 


.89113 
.89142 
-89171 
-89200 
-89229 


8-18553 
8-20999 
8-23459 
8-25931 
8-28417 


45 
46 
47 
48 
49 


50 

51 
52 
53 
54 


•84069 
•84098 
.84127 
.84155 
.84184 


5.27719 
5.28853 
5.29991 
5-31133 
5.32279 
5.33429 
5.34584 
5-35743 
5-36906 
5.38073 


-85795 
.85823 
.85852 
.85881 
-85910 


6.03962 
6.05392 
6.06828 
6-08269 
6.09717 


.87524 
.87553 
.87582 
.87611 
•87640 


7-01565 
7-03423 
7-05291 
7-07167 
7-09052 


-89258 
-89287 
-89316 
-89345 
.89374 


8-30917 
8-33430 
8-35957 
8-38497 
8-41052 


50 

51 

52 
53 
54 


55 
56 
57 
58 
59 


.84213 
.84242 
•84270 
•84299 
•84328 


-85939 
-85967 
-85996 
-86025 
.86054 


6-11171 
6-12630 
6.14096 
6.15568 
6-17046 


•87669 
.87698 
.87726 
•87755 
.87784 


7.10946 
7.12849 
7.14760 
7-16681 
7.18612 


.89403 
.89431 
.89460 
-89489 
.89518 


8-43620 
8-46203 
8-48800 
8-51411 
8.54037 


55 
56 
57 
58 
59 


60 


.84357 


5-39245 


.86083 


6.18530 


-87813 


7.20551 


•89547 


8-56677 


60 



738 



TABLE X.— NATURAL 

84" 


VERSED SINES AND 
85° 


EXTERNAL SECANTS. 

86° 


/ 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


9 




1 

2 
3 
4 


.89547 
.89576 
.89605 
.89634 
• 89663 


8.56677 
8.59332 
8.62002 
8.64687 
8.67387 




.91284 
.91313 

91342 
91371 
91400 


10.47371 
10.51199 
10.55052 
10.58932 
10.62837 




93024 
.93053 
.93082 
.93111 
.93140 


13.33559 
13.39547 
13.45586 
13-51676 
13.57817 




1 
2 
3 
4 


5 
6 
7 
8 
9 

LO 

11 
12 
13 
14 
15 
16 
17 
18 
19 


.89692 
.89721 
.89750 
•89779 
•89808 


8.70103 
8.72833 
8.75579 
8.78341 
8.81119 




91429 
91458 
91487 
91516 
91545 


10.66769 
10.70728 
10.74714 
10.78727 
10.82768 




.93169 
•93198 
.93227 
.93257 
.93286 


13.64011 
13.70258 
13.76558 
13.82913 
13.89323 


5 
6 
7 
8 
9 


•89836 
.89865 
•89894 
•89923 
.89952 


8.83912 
8.86722 
8.89547 
8.92389 
8.95248 




91574 
91603 
91632 
91661 
91690 


10.86837 
10.90934 
10.95060 
10.99214 
11.03397 




.93315 
.93344 
.93373 
.93402 
93431 


13.95788 
14.02310 
14.08890 
14.15527 
14.22223 


10 

11 
12 
13 
14 


•89981 
•90010 
•90039 
•90068 
.90097 


8.98123 
9^01015 
9.03923 
9.06849 
9.09792 




91719 
91748 
91777 
91806 
91835 


31.07610 
11.11852 
11.16125 
11.20427 
11.24761 




93460 
93489 
93518 
93547 
93576 


14.28979 
14.35795 
14.42672 
14.49611 
14.56614 


15 
16 
17 
18 
19 


JO 

21 
22 
^3 
24 


•90126 
•90155 
•90184 
•90213 
•90242 


9.12752 
9.15730 
9.18725 
9.21739 
9.24770 




91864 
91893 
91922 
91951 
91980 


11.29125 
11.33521 
11.37948 
11.42408 
11.46900 




93605 
93634 
93663 
93692 
93721 


14.63679 
14^70810 
14^78005 
14^85268 
14^92597 


20 

21 

22 
23 
24 


25 
26 
27 
28 
29 


•90271 
.90300 
•90329 
•90358 
•90386 


9.27819 
9.30887 
9.33973 
9.37077 
9.40201 




92009 
92038 
92067 
92096 
92125 


11.51424 
11.55982 
11.60572 
11.65197 
11.69856 




93750 
93779 
93808 
93837 
93866 


14^99995 
15-07462 
15-14999 
15^22607 
15.30287 


25 
26 
27 
28 
29 


^0 

31 
32 
33 
34 


•90415 
.90444 
.90473 
.90502 
•90531 


9.43343 
9.46505 
9.49685 
9.52886 
9.56106 




92154 
92183 
92212 - 
92241 
92270 


11.74550 
11.79278 
11.84042 
11.88841 
11.93677 




93895 
93924 
93953 
93982 
94011 


15.38041 
15.45869 
15.53772 
15.61751 
15.69808 


30 

31 
32 
33 
34 


85 
36 
37 
38 
^9 


•90560 
•90589 
•90618 
•90647 
•90676 


9.59346 
9.62605 
9.65885 
9.69186 
9.72507 




92299 
92328 
92357 
92386 
92415 


11.98549 
12.03458 
12.08040 
12.13388 
12.18411 




94040 
94069 
94098 
94127 
94156 


15.77944 
15.86159 
15.94456 
16.02835 
16.11297 


35 
36 
37 
38 
39 


to 

41 
42 
43 
44 


•90705 
•90734 
•90763 
•90792 
•90821 


9.75849 
9.79212 
9 •82593 
9.86001 
9 . 89428 




92444 
92473 
92502 
92531 
92560 


12.23472 
12.28572 
12.33712 
12.38891 
12.44112 




94186 
94215 
94244 
94273 
94302 


16.19843 
16.28476 
16.37196 
16.46005 
16.54903 


40 

41 
42 
43 
44 


45 
66 
(47 
^8 
49 


.90850 
.90879 
.90908 
.90937 
.90966 


9.92877 

9.96348 

9.99841 

10.03356 

10.06894 




92589 
92618 
92647 
92676 
92705 


12.49373 
12.54676 
12.60021 
12.65408 
12.70838 




94331 
94360 
94389 
94418 
94447 


16.63893 
16.72975 
16.82152 
16.91424 
17.00794 


45 
46 
47 
48 
49 


50 

63 

p4 


•90995 
•91024 
.91053 
.91082 
.91111 


10.10455 
10.14039 
10.17646 
10.21277 
10.24932 




92734 
92763 
92792 
92821 
92850 


12.76312 
12.81829 
12.87391 
12.92999 
12.98651 




94476 
94505 
94534 
94563 
94592 


17.10262 
17.19830 
17.29501 
17.39274 
17.49153 


50 

51 

52 
53 
54 


*55 
j56 
(57 
(58 

1^ 


.91140 
.91169 
.91197 
.91228 
.91255 

.91284 


10.28610 
10.32313 
10.36040 
10.39792 
10^43569 




92879 
92908 
92937 
92966 
92995 . 


13.04350 
13.10096 
13.15889 
13.21730 
13.27620 




94621 
94650 
94679 
94708 
94737 


17.59139 
17.69233 
17.79438 
17.89755 
18.00185 


55 
56 
57 
58 
59 


10.47371 




93024 


13.33559 




94766 


18.10732 


60 



739 



TABLE X.— NATURAL 


VERSED SINES AND EXTERNAL SECANTS. 




87° 




88° 




89° 




1 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


Vers. 


Ex. sec. 


/ 


"o" 


.94766 


18 


10732 


.96510 


27 


65371 


.98255 


56.29869 


~ 


1 


.94795 


18 


.21397 


.96539 


27 


89440 


.98284 


57.26976 


1 


2 


.94825 


18 


.32182 


.96568 


28 


13917 


.98313 


58.27431 


2 


3 


.94854 


18 


43088 


.96597 


28 


38812 


.98342 


59.31411 


3 


4 


.94883 


18 


54119 


.96626 


28 


64137 


.98371 


60.39105 


4 


5 


.94912 


18 


65275 


.96655 


28 


89903 


•98400 


61.50715 


5 


6 


.94941 


18 


76560 


.96684 


29 


16120 


.98429 


62.66460 


6 


7 


.94970 


18 


87976 


.96714 


29 


42802 


.98458 


63.86572 


7 


8 


.94999 


18 


99524 


.96743 


29 


69960 


.98487 


65.11304 


8 


9 


.95028 


19 


11208 


.96772 


29 


97607 


.98517 


66.40927 


9 


10 


.95057 


19 


23028 


.96801 


30 


25758 


.98546 


67.75736 


10 


11 


.95086 


19 


34989 


.96830 


30 


54425 


.98575 


69.16047 


11 


12 


.95115 


19 


47093 


.96859 


30 


83623 


.98604 


70.62285 


12 


13 


.95144 


19 


59341 


.96888 


31 


13366 


.98633 


72.14583 


13 


14 


.95173 


19 


71737 


.96917. 


31 


43671 


.98662 


73.73586 


14 


15 


.95202 


19 


84283 


.96946 


31 


74554 


.98691 


75.39655 


15 


16 


.95231 


19 


96982 


.96975 


32 


06030 


.98720 


77.13274 


16 


17 


.95260 


20 


09838 


.97004 


32 


38118 


.98749 


78.94968 


17 


18 


.95289 


20 


22852 


.97033 


32 


70835 


.98778 


80.85315 


18 


I9_ 


.95318 


20 


36027 


.97062 


33 


04199 


.98807 


82.84947 


19 


^0 


.95347 


20 


49368 


.97092 


33 


38232 


.98836 


84.94561 


20 


21 


.95377 


20 


62876 


.97121 


33 


72952 


.S8866 


87.14924 


21 


22 


.95406 


20 


76555 


.97150 


34 


08380 


.98895 


89.46886 


22 


23 


.95435 


20 


90409 


.97179 


34 


44539 


.98924 


91.91387 


23 


24 


.95464 


21 


04440 


.97208 


34 


81452 


.98953 


94.49471 


24 
25 


25 


.95493 


21. 


18653 


.97237 


35 


19141 


.98982 


97.22303 


26 


.95522 


21. 


33050 


.97266 


35 


57633 


.99011 


100.1119 


26 


27 


.95551 


21. 


47635 


.97295 


35 


96953 


.99040 


103.1757 


27 


28 


.95580 


21 


62413 


.97324 


36 


37127 


.99069 


106.4311 


28 


29 


.95609 


21 


77386 


.97353 


36 


78185 


.99098 


109.8966 


29 
30 


30 


.95638 


21 


92559 


.97382 


37 


20155 


.99127 


113.5930 


31 


.95667 


22 


07935 


.97411 


37 


63068 


.99156 


117.5444 


31 


32 


.95696 


22 


23520 


.97440 


38 


06957 


.99186 


121.7780 


32 


33 


.95725 


22 


39316 


.97470 


38 


51855 


.99215 


126.3253 


33 


34 


.95754 


22 


55328 


.97499 


38 


97797 


.99244 


131.2223 


34 
35 


35 


.95783 


22 


71563 


.97528 


39 


44820 


.99278 


136.5111 


36 


.95812 


22 


88022 


.97557 


39 


92963 


.99302 


142.2406 


36 


37 


.95842 


23 


04712 


.97586 


40 


42266 


.99331 


148.4684 


37 


38 


.95871 


23 


21637 


.97615 


40 


92772 


.99360 


155.2623 


38 


39. 


.95900 


23 


38802 


.97644 


41 


44525 


.99389 


162.7033 


39 


40 


.95929 


23 


56212 


.97673 


41 


97571 


.99418 


170.8883 


40 


41 


.95958 


23 


73873 


.97702 


42 


51961 


.99447 


179.9350 


41 


42 


.95987 


23 


91790 


.97731 


43 


07746 


.99476 


189.9868 


42 


43 


.96016 


24 


09969 


.97760 


43 


64980 


.99505 


201.2212 


43 


44 


.96045 


24 


28414 


.97789 


44 


23720 


.99535 


213.8600 


44 


45 


.96074 


24 


47134 


.97819 


44 


84026 


.99564 


228.1839 


45 


46 


.96103 


24 


66132 


.97848 


45 


45963 


.99593 


244.5540 


46 


47 


.96132 


24 


85417 


.97877 


46 


09596 


.99622 


263.4427 


47 


48 


.96161 


25 


04994 


.97906 


46 


74997 


.99651 


285.4795 


48 


49 


•96190 


25 


24869 


.97935 


47 
48 


42241 


.99680 


311.5230 


49 


60 


.96219 


25 


45051 


.97964 


11406 


.99709 


342.7752 


50 


51 


.96248 


25 


65546 


.97993 


48 


82576 


.99738 


380.9723 


51 


52 


.96277 


25 


.86360 


.98022 


49 


55840 


.99767 


428.7187 


52 


53 


.96307 


26 


.07503 


.98051 


50 


31290 


.99796 


490.1070 


53 


54 


.96336 


26 


.28981 


.98080 


51 


09027 


.99825 


571.9581 


54 
55 


55 


.96365 


26 


.50804 


.98109 


51 


89156 


.99855 


686.5496 


56 


.96394 


26 


.72978 


.98138 


52 


71790 


.99884 


858.4369 


56 


57 


.96423 


26 


.95513 


.98168 


53 


57046 


.99913 


1144.916 


57 


58 


.96452 


27 


.18417 


.98197 


54 


45053 


.99942 


1717.874 


58 


J^ 


.96481 


27 


.41700 


.98226 


55 


35946 


.99971 


3436.747 


59 


60 


.96510 


27 


.65371 


.98255 


56 


29869 


1.00000 


Infinite 


60 



740 



TABLE XI.— REDUCTION OF BAROMETER READING TO 32° F. 















[nches. 












Temp. 
























O 
Fahr. 


26-0 


26.5 


27.0 


27.5 


28.0 


28.5 


29.0 


29.5 


300 


30.5 


310 


45 


-.039 


-.039 


-.040 


-.041 


-.042 


042 


-.043 


-.044 


-.045 


-.045 


-.046 


46 


.041 


.042 


.043 


.043 


.044 


045 


•046 


•046 


.047 


.048 


.049 


47 


.043 


.044 


.045 


.046 


.047 


.048 


.048 


.049 


.050 


.051 


.052 


48 


.046 


•047 


.047 


.048 


.049 


.050 


.051 


.052 


.053 


.053 


.054 


49 


.048 


.049 


.050 


.051 


.052 


.052 


.054 


.054 


.055 


.056 


.057 


50 


.050 


.051 


.052 


.053 


.054 


.055 


.056 


.057 


.058 


.059 


.060 


51 


.053 


.054 


.055 


.056 


.057 


.058 


.059 


.060 


.061 


.062 


.063 


52 


•055 


.056 


057 


.058 


.059 


.060 


.061 


.062 


.064 


.065 


.066 


53 


.057 


•058 


.060 


.061 


.062 


.063 


.064 


.065 


.066 


.067 


.068 


54 


.060 


.061 


.062 


.063 


.064 


.065 


.067 


.068 


.069 


.070 


.071 


55 


.062 


•063 


.064 


.065 


.066 


.068 


.069 


.070 


.071 


.073 


.074 


56 


.064 


•065 


.067 


.068 


.069 


.070 


.072 


.073 


.074 


.075 


.077 


57 


.067 


•068 


.069 


.070 


.072 


.073 


.075 


.076 


.077 


.078 


.080 


58 


.069 


.070 


.071 


.073 


.074 


.076 


.077 


.078 


.080 


.081 


.082 


59 


.072 


.073 


.074 


.075 


.077 


.078 


.080 


.081 


.083 


.084 


.085 


60 


.074 


.076 


.077 


.078 


.079 


.081 


.082 


.084 


.085 


.086 


.088 


61 


.076 


.077 


.079 


.080 


.082 


.083 


.085 


.086 


.088 


.089 


.091 


62 


.079 


.080 


.082 


.083 


.085 


.086 


.088 


.089 


.091 


.092 


.094 


63 


.081 


•082 


.084 


.085 


.087 


.088 


.090 


.091 


.093 


.095 


.096 


64 


.083 


.085 


.086 


.088 


.090 


.091 


.093 


.094 


.096 


.097 


.099 


65 


.086 


.087 


.089 


.090 


.092 


.093 


.095 


.097 


.099 


.100 


.102 


66 


.088 


.089 


.091 


.093 


.095 


.096 


.098 


.099 


.101 


.103 


.105 


67 


.090 


.092 


.094 


•095 


.097 


.099 


.101 


.102 


.104 


.106 


.108 


68 


.093 


.094 


.096 


.098 


.100 


.101 


.103 


.105 


.107 


.108 


.110 


69 


.095 


.097 


.099 


.100 


.102 


.104 


.106 


.107 


.110 


.111 


.113^ 


70 


.097 
.100 


.099 
.101 


.101 
.103 


.103 
.105 


.105 


.106 


.109 
.111 


.110 
.113 


.112 
.115 


.114 
.117 


.116 


71 


• 107 


.109 


.119 


72 


.102 


.104 


.106 


.108 


.110 


.112 


.114 


.116 


.118 


.120 


.122 


73 


.104 


.106 


.108 


.110 


.112 


.114 


.116 


.118 


.120 


.122 


.124 


74 


.107 


• 109 


.111 


.113 


.115 


.117 


.119 


.121 


.123 


.125 


.127 


75 


.109 


•111 


.113 


.115 


.117 


.119 


.122 


.124 


.126 


.128 


.130 


76 


.111 


• 113 


.116 


.118 


.120 


.122 


.124 


.126 


.128 


.130 


.133 


77 


.114 


.116 


.118 


.120 


.122 


.124 


.127 


.129 


.131 


.133 


.136 


78 


.116 


.118 


.120 


.122 


.125 


.127 


.129 


.131 


.134 


.136 


.138 


79 


.118 


.120 


.123 


.125 


.127 


.129 


.132 


.134 


.137 


.139 


.141 


80 


.121 


.123 


.125 


.127 


.130 


.132 


.135 


.137 


.139 


.141 


.144 


81 


.123 


.125 


.128 


.130 


.132 


.134 


.137 


.139 


.142 


.144 


.147 


82 


.125 


.128 


.130 


.132 


.135 


.137 


.140 


.142 


.145 


.147 


.149 


83 


• 128 


.130 


.133 


.135 


.138 


.140 


.142 


.145 


.147 


.149 


.152 


84 


.130 


• 132 


.135 


.138 


.140 


.142 


.145 


.147 


.150 


• 152 


.155 


85 


.132 


• 134 


.137 


.140 


.143 


.145 


.148 


.150 


.153 


.155 


.158 


86 


.135 


.137 


.140 


.142 


• 145 


.148 


.150 


.153 


.155 


.158 


.161 


87 


.137 


.139 


.142 


.144 


.148 


.150 


.153 


.155 


• 158 


.161 


•163 


88 


.139 


.142 


.145 


.147 


.150 


.152 


.155 


.158 


.161 


.163 


.166 


89 


.142 


.144 


.147 


.150 


.153 


.155 


.158 


.161 


.164 


.166 


.169 


90 


.144 


.147 


.150 


.153 


.155 


.158 


.161 


.164 


.166 


.169 


.172 


. 91 


-.146 


-.149 


-.152 


-.155 


-.158 


-.160 


-.163 


-.166 


-.169 


-.172 


-.175 



741 



TABLE XII.—BAROMETRIC ELEVATIONS.* 



Inches. 



20.0 
20.1 
20.2 
20.3 
20.4 
20.5 
20.6 
20.7 
20.8 
20.9 
21.0 
21.1 
21.2 
21.3 
21.4 
21.5 
21.6 
21.7 
21.8 
21.9 
22.0 
22.1 
22.2 
22.3 
22.4 
22.5 
22.6 
22.7 
22.8 
22.9 
23. 
23.1 
23.2 
23.3 
23.4 
23. 5 
23.6 
23.7 



Feet. 

11,047 
10,911 
10,776 
10,642 
10,508 
10,375 
10,242 
10,110 
9,979 
9,848 
9,718 
9,589 
9,460 
9,332 
9.204 
9,077 
8,951 
8,825 
8,700 
8,575 
8,451 
8,327 
8,204 
8,082 
7,960 
7,838 
7,717 
7,597 
7,477 
7-358 
7,239 
7,121 
7,004 
6,887 
6,770 
6,554 
6,538 
6,423 



Diff . for 
.01. 



Feet. 



-13 


6 


13 


5 


13 


4 


13 


4 


13 


3 


13 


3 


13 


2 


13 


1 


13 


1 


13 





12 


9 


12 


9 


12 


8 


12 


8 


12 


7 


12 


6 


12 


6 


12 


5 


12 


5 


12 


4 


12 


4 


12 


3 


12 


2 


12 


2 


12 


2 


12 


1 


12 





12 







9 




9 




8 




7 




7 




7 




6 




6 


— 11 


5 



Inches. 

23 
23 
23 
24 
24 
24 
24 
24 
24 
24 
24 
24 
24 
25 
25 
25 
25 
25 
25 
25 
25 
25 
25 
26 
26 
26 
26 



26 
26 
26 
26 
26 
27 
27 
21 
27 
27 



Feet. 

6^423 
6,308 
6,194 
6,080 
5,967 
5,854 
5,741 
5,629 
5,518 
5.407 
5,296 
5,186 
5,077 
4,968 
4,859 
4,751 
4,643 
4,535 
4,428 
4,321 
4,215 
4.109 
4,004 
3 899 
3,794 
3 690 
3,586 
3,483 
3,380 
3.277 
3,175 
3,073 
2 972 
2.871 
2 770 
2,670 
2,570 
2,470 



Diff. for 
.01. 



Feet. 



11 


5 


11 


4 


11 


4 


11 


3 


11 


3 


11 


3 


11 


2 


11 


1 


11 


1 


11 


1 


11 





10 


9 


10 


9 


10 


9 


10 


8 


10 


8 


10 


8 


10 


7 


10 


7 


10 


6 


10 


6 


10 


5 


10 


5 


10 


4 


10 


4 


10 


4 


10 


3 


10 


3 


10 


3 


10 


2 


10 


2 


10 


1 


10 


1 


10 


1 


10 





10 





10 






B 




A 


Inches. 


Feet. 


27.4 


2,470 


27 


5 


2,371 


27 


6 


2,272 


27 


7 


2,173 


27 


8 


2,075 


27 


9 


1,977 


28 





1,880 


28 


1 


1,783 


28 


2 


1 686 


28 


3 


1,589 


28 


4 


1,493 


28 


5 


1,397 


28 


6 


1,302 


28 


7 


1,207 


28 


8 


1,112 


28 


9 


1,018 


29 





924 


29 


1 


830 


29 


2 


736 


29 


3 


643 


29 


4 


550 


29 


5 


458 


29 


6 


366 


29 


7 


274 


29 


8 


182' 


29 


9 


91 


30 








30 


1 


-91 


30 


2 


181 


30 


3 


271 


30 


4 


361 


30 


5 


451 


30 


6 


540 


30 


7 


629 


30 


8 


717 


30 


9 


805 


31. 


-893 



* Compiled from Report of U. S. C. & G. Survey for 1881, App. 10 Table XI. 



TABLE XIIL— COEFFICIENTS FOR CORRECTIONS FOR TEMPERATURE 
AND HUMIDITY.* 



t^-t' 



0° 
10 
20 
30 
40 
50 
60 



.1024 
.0915 
.0806 
.0698 
.0592 
.0486 
.0380 



Diff. for 
1°. 



t^-t' 



60° 

70 

80 

90 

100 

110 

120 



-.0380 
• 0273 
.0166 

-.0058 

+ .0049 
.0156 

+ .0262 



Diff. for 
1". 



10 
10 
10 
10 
10 



10-6 



t^t' 


C 


120° 


+ .0262 


130 


.0368 


140 


.0472 


150 


.0575 


160 


.0677 


170 


.0779 


180 


+ .0879 



* Compiled from Report of U. S. C. & G. Survey for 1881, App. 10, Tables I, IV. 

742 



TABLE XXX.— USEFUL TRIGONOMETEICAL FORMULtE. 



/I — cos 2a 



1 _ tan a _ / ] 

cosec a ~ Vi + tan^ a V ^ Vl + cot2 a 

= cos a tan a == v 1 — cos^ a = 2 sin ^a cos ^a 

1 4- cos a 2 tan ^a ^ . 

= i-i — ■ ""TTT — ^71— = vers a cot ^a. 

cot ^a 1 + tan^ ^a 



cos a = 



sec a Vi4-cot2 a Vl4-tan2 a 
= 1 — vers a = sin a cot a = V^l— sin^ a = 2 cos^ ^a — 1 
= sin a cot ^a— 1 ^cos- ^a — sin2 ^a= 1 — 2 sin^ ^a. 
1 sin a sec a 1 



cot a cos a cosec a V'cosec^ a — 1 

= vers 2a cosec 2o = cot a — 2 cot 2a = sin a sec a 

sin 2a , , o ^ o 

= -— = exsec a cot ^a = 3xsec 2o cot 2a. 

1 + cos 2a 

_ 1 _ cos g _ sin 2a __ 1 + cos 2a 
tan a sin a 1 — cos 2a sin 2a 

= V cosec2 a — 1 = cot \a — cosec a. 

= 1 — cos a = sin a tan \a = 2 sin^ ^a = cos a exsec a„ 

= sec a — 1 = tan a tan +a = vers a sec a. 



sin ^a 

cos ^a = 

tan ^a 

cot ^a -- 
vers i^a = 
exsec ^a = 



/vers a 



^ vers a _ sin a _ vers a cos ^a 
2 cos ia sin a 



/ 1 + cos g sin a _sin a sin ^a 

I' 2 2 sin ^a vers a 



= vers a cosec a = cosec a — cot o 
l + cos g 



sm g 

^ 1 - ^+(l+cosg), 
1 



= cosec a + cot a = - 



tan a 
1+sec a ' 

tan g 



exsec a cosec a— cot o 



VA(i + cosa) 



743 



TABLE XXX.— USEFUL TRIGONOMETRICAL FORMULA. 



. r. o • 2 tan a 

sm 2a =2 sin a cos a = 



1 + tan2 a 

cos 2a =cos2 a— sin2 a=l — 2sin2a=2 cos^a— 1 
1 — tan2 a 



tan 2a 



1 + tan2 a 

2 tan a 
1 — tan2 a* 



cot 2a =i cot a-^ tan a = ^^^' ^""^ ^l-tan^ g 
2 cot a 2 tan a " 

vers 2a = 2 sin2 a = 1 - cos 2a = 2 sin a cos a tan a, 

exsec2a = -i^^2a___2tan2a _ 2 sin2 a 



cot a 1 - tan2 a ~ 1 -2 sin2 ^ • 
sin (a ± 6) =sin a cos h ± cos a sin 6. 
cos (a ± 6) = cos a cos 6 T sin a sin 6. 
sin a + sin 6 =2 sin i(a + 6) cos i^(a — 6). 
sin a— sin 6 =2 sin i-(a — 6) oos ^(a + 6). 
cos a + cos 6 = 2 cos i(a + 6) cos i(a — 6). 
cos a — cos 6= —2 sin i(a + 6) sin ^(a — 6). 



Call the sides of any triangle A,B, C, and the opposite angles a, &.. 
andc. Calls = iU+5 + C). 

tan i(a - 6) = . p tan ^(a + 6) = . p Cot ^c. 
cosi(a — 6) ^sini(a — 6) 



^^a^^i/^-^ 



_ (s-5)(s-C) 
sm f - - 



cosi^( 



/ s{s-A ) 

2{s-B){s-C) 
BC 



Ktq2. -Vs(s-^)(s-5)(s-C)=A2 



744 



sin b sin c 
2 sin a 



TABLE XXXI.— USEFUL FORMULA AND CONSTANTS. 



Circumference of a circle (radius = r) = 2nr, 

Area of a circle = nr^. 

Area of sector (length of arc = = ilr, 

" " " (angle of arc = a°) = „* Trr^. 

Area of segment (chord = v, mid. ord. = m) = %cm (approx.). 

Area of a circle to radius 1 1 

I 
Circumference of a circle to diameter 1 \ = rr = 3.1415927 

Surface of a sphere to diameter 1 J 

Volume of a sphere to radius 1 = 47r -f- 3 = 4.1887902 

r degrees = 57.2957795 

I 
Arc equal to radius expressed in •{ minutes = 3437.7467708 

I 
L seconds = 206264.8062471 

Length of arc of 1°, radius unity 0.01745329 

Sine of one second = 0.0000048481 

Cubic inches in United States standard gallon = 231 

Weight of one cubic foot of water at maximum density (therm. 

39°.8 F., barom. 30") ._^ 62.379 

Weight of one cubic foot of water at ordinary temperature (therm. 

62° F.) 62.321 

Acceleration due to gravity at latitude of New York in feet per 

square second 32.15945 

Feet in one metre 3.280869 

Metres in one foot 0.304797 

745 



Logarithm. 



0.4971499 

0.622 0886 
1.7581226 
3.536 2739 
5.314 4251 
8.2418774 
4.6855749 
2.363 6120 

1.795 0384 

1.794 6349 

1.507 3086 
0.515 9889 
9.484 0111 





TABLE XXXII— SQUARES, CUBES, 


SQUARE ROOTS, 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


1 


1 


1 


1.0000000 


1.0000000 


1.000000000 


2 


4 


8 


1.4142136 


1.2599210 


.500000000 


3 


9 


27 


1.7320508 


1.4422496 


.333333333 


4 


16 


64 


2.0000000 


1.5874011 


.250000000 


5 


25 


125 


2.2360680 


1.7099759 


.200000000 


6 


36 


216 


2.4494897 


1.8171206 


.166666667 


7 


49 


343 


2.6457513 


1.9129312 


.142857143 


8 


64 


512 


2.8284271 


2.0000000 


.125000000 


9 


81 


729 


3.0000000 


2.0800837 


.111111111 


10 


100 


1000 


3.1622777 


2.1544347 


.100000000 


11 


121 


1331 


3.3166248 


2.2239801 


.090909091 


12 


144 


1728 


3.4641016 


2.2894286 


.083333333 


13 


169 


2197 


3.6055513 


2.3513347 


.076923077 


14 


196 


2744 


3.7416574 


2.4101422 


.071428571 


15 


225 


3375 


3-8729833 


2.4662121 


.066666667 


16 


256 


4096 


4.0000000 


2.5198421 


.062500000 


17 


289 


4913 


4.1231056 


2.5712816 


.058823529 


18 


324 


5832 


4.2426407 


2.6207414 


.055555556 


19 


361 


6859 


4.3588989 


2.6684016 


.052631579 


20 


400 


8000 


4.4721360 


2.7144177 


.050000000 


21 


441 


9261 


4.5825757 


2.7589243 


.047619048 


22 


484 


10648 


4.6904158 


2.8020393 


.045454545 


23 


529 


12167 


4.7958315 


2.8438670 


.043478261 


24 


576 


13824 


4.8989795 


2.8844991 


.041666667 


25 


625 


15625 


5.0000000 


2.9240177 


.040000000 


26 


676 


17576 


5.0990195 


2.9624960 


.038461538 


27 


729 


19683 


5.1961524 


3.0000000 


.037037037 


28 


784 


21952 


5.2915028 


3.0365889 


.035714286 


29 


841 


24389 


5.3851648 


3.0723168 


.034482759 


30 


900 


27000 


5.4772256 


3.1072325 


.033333333 


31 


961 


29791 


5.5677644 


3.1413806 


.032258065 


32 


1024 


82768 


5.6568542 


3.1748021 


.031250000 


33 


1089 


35937 


5.7445626 


3.2075343 


.030303030 


34 


1156 


39304 


5.8309519 


3.2396118 


.029411765 


35 


1225 


42873 


5.9160798 


3.2710663 


.028571429 


36 


1296 


46656 


6.0000000 


3.3019272 


.027777778 


37 


1369 


50653 


6.0827625 


3.3322218 


.027027027 


38 


1444 


54872 


6.1644140 


3.3619754 


.026315789 


39 


1521 


59319 


6.2449980 


3.3912114 


.025641026 


40 


1600 


64000 


6.3245553 


3.4199519 


.025000000 


41 


1681 


68921 


6.4031242 


3.4482172 


.024390244 


42 


1764 


74088 


6.4807407 


3.4760266 


.023809524 


43 


1849 


79507 


6.5574385 


3.5033981 


.023255814 


44 


1936 


85184 


6.6332496 


3.5303483 


.022727273 


45 


2025 


91125 


6.7082039 


3.5568933 


.022222222 


46 


2116 


97336 


6.7823300 


3.5830479 


.021739130 


47 


2209 


103823 


6.8556546 


3.6088261 


.021276600 


48 


2304 


110592 


6.9282032 


3.6342411 


.020833333 


49 


2401 


117649 


7.0000000 


3.6593057 


.020408163 


50 


2500 


125000 


7.0710678 


3.6840314 


.020000000 


51 


2601 


132651 


7.1414284 


3.7084298 


.019607843 


52 


2704 


140608 


7.2111026 


3.7325111 


.019230769 


53 


2809 


148877 


7.2801099 


3.7562858 


.018867925 


54 


2916 


157464 


7.3484692 


3.7797631 


.618518519 


55 


3025 


166375 


7.4161985 


3.8029525 


.018181818 


56 


3136 


175616 


7.4833148 


3.8258624 


.017857143 


57 


3249 


185193 


7.5498344 


3.8485011 


.017543860 


58 


3364 


195112 


7.6157731 


3.8708766 


.017241379 


59 


3481 


205379 


7.6811457 


3.8929965 


.016949153 


<oO 


3600 


216000 


7.7459667 


3.9148676 


.016666667 



746 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


61 
62 
63 
64 
65 


3721 
3844 
3969 
4096 
4225 


226981 
238328 
250047 
262144 
274625 


7.8102497 
7.8740079 
7.9372539 
8.0000000 
8.0622577 


3.9364972 
3.9578915 
3-9790571 
4.0000000 
4-0207256 


.016393443 
.016129032 
.015873016 
.015625000 
.015384615 


66 
67 
68 
69 
70 


4356 
4489 
4624 
4761 
4900 


287496 
300763 
314432 
328509 
343000 


8.1240384 
8.1853528 
8.2462113 
8.3066239 
8-3666003 


4.0412401 
.4.0615480 
4.0816551 
4.1015661 
4-1212853 


.015151515 
.014925373 
.014705882 
.014492754 
.014285714 


71 
72 
73 
74 
75 
76 
77 
78 
79 
80 


5041 
5184 
5329 
5476 
5625 


357911 
373248 
389017 
405224 
421875 


8.4261498 
8.4852814 
8.5440037 
8.6023253 
8.6602540 


4.1408178 
4.1601676 
4.1793390 
4.1983364 
4-2171633 


.014084507 
.013888889 
.013698630 
.013513514 
.013333333 


5776 
5929 
6084 
6241 
6400 


438976 
456533 
474552 
493039 
512000 


8.7177979 
8.7749644 
8.8317609 
8.8881944 
8.9442719 


4.2358236 
4.2543210 
4-2726586 
4.2908404 
4-3088695 


.013157895 
.012987013 
.012820513 
.012658228 
.012500000 


81 

ff2 

1 83 

' 84 

• 85 


6561 
6724 
6889 
7056 
7225 


531441 
551368 
571787 
592704 
614125 


9.0000000 
9.0553851 
9.1104336 
9.1651514 
9.2195445 


4.3267487 
4.3444815 
4.3620707 
4.3795191 
4-3968296 


.012345679 
.012195122 
.012048193 
.011904762 
.011764706 


! 86 
87 
, 88 
' 89 
j 90 


7396 
7569 
7744 
7921 
8100 


636056 
658503 
681472 
704969 
729000 


9.2736185 
9.3273791 
9.3808315 
9.4339811 
9.4868330 


4-4140049 
4-4310476 
4.4479602 
4.4647451 
4-4814047 


.011627907 
.011494253 
.011363636 
.011235955 
.011111111 


' 92 

i 93 

; 94 

95 


8281 
8464 
8649 
8836 
9025 


753571 
778688 
804357 
830584 
857375 


9.5393920 
9.5916630 
^76436508 
9.6953597 
9.7467943 


4.4979414 
4.5143574 
4.5306549 
4.5468359 
4.5629026 


.010989011 
.010869565 
.010752688 
.010638298 
.010526316 


' 96 

97 

1 98 

1 99 

100 


9216 
9409 
9604 
9801 
10000 


884736 
912673 
941192 
970299 
1000000 


9.7979590 
9.8488578 
9.8994949 
9.9498744 
10.0000000 


4.5788570 
4.5947009 
4.6104363 
4.6260650 
4-6415888 


.010416667 
.010309278 
.010204082 
.010101010 
-010000000 


; 101 

\ 102 
^ 103 
, 104 
, 105 


10201 
10404 
10609 
10816 
11025 


1030301 
1061208 
1092727 
1124864 
1157625 


10.0498756 
10.0995049 
10.1488916 
10.1980390 
10.2469508 


4.6570095 
4.6723287 
4.6875482 
4.7026694 
4-7176940 


.009900990 
.009803922 
.009708738 
.009615385 
.009523810 


j 106 

1 107 

1 108 

109 

110 


11236 
11449 
11664 
11881 
12100 


1191016 
1225043 
1259712 
1295029 
1331000 


10.2956301 
10.3440804 
10.3923048 
10.4403065 
10.4880885 


4.7326235 
4.7474594 
4.7622032 
4.7768562 
4.7914199 


.009433962 
.009345794 
.009259259 
.009174312 
.009090909 


111 
{ 112 
113 
114 
115 


12321 
12544 
12769 
12996 
13225 


1367631 
1404928 
1442897 
1481544 
1520875 


10.5356538 
10.5830052 
10.6301458 
10.6770783 
10.7238053 


4.8058955 
4.8202845 
4.8345881 
4.8488076 
4.8629442 


.009009009 
.008928571 
.008849558 
.008771930 
.008695652 


116 
I 117 
' 118 
' 119 

120 


13456 
13689 
13924 
14161 
14400 


1560896 
1601613 
1643032 
1685159 
1728000 


10.7703296 
10.8166538 
10.8627805 
10.9087121 
10.9544512 


4.8769990 
4.8909732 
4.9048681 
4.9186847 
4.9324242 


.008620690 
.008547009 
.008474576 
.008403361 
.008333333 








747 









TABLE XXXII —SQUARES, CUBES, 


SQUARE ROOTS, 


No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


121 
122 
123 
124 
125 


14641 
14884 
15129 
15376 
15625 


1771561 
1815848 
1860867 
1906624 
1953125 


11 
11 
11 
11 
11 


0000000 
.0453610 
.0905365 
.1355287 
.1803399 


4 
4 
4 
4 
5 


9460874 
9596757 

.9731898 
9866310 

■0000000 




.008264463 
.008196721 
.008130081 
.008064516 
.008000000 


126 
127 
128 
129 
130 


15876 
16129 
16384 
16641 
16900 


2000376 
2048383 
2097152 
2146689 
2197000 


11 
11 
11 
11 
11 


.2249722 
.2694277 
.3137085 
.3578167 
.4017543 


5 
5 
5 
5 
5 


•0132979 
.0265257 
•0396842 
.0527743 
•0657970 




.007936508 
.007874016 
.007812500 
.007751938 
.007692308 


131 
132 
133 
134 
135 


17161 
17424 
17689 
17956 
18225 


2248091 
2299968 
2352637 
2406104 
2460375 


11 
11 
11 
11 
11 


4455231 
4891253 
5325626 
5758369 
6189500 


5 
5 
5 
5 
5 


.0787531 
.0916434 
•1044687 
.1172299 
.1299278 




•007633588 
.007575758 
.007518797 
.007462687 
.007407407 


136 
137 
138 
139 
140 


18496 
18769 
19044 
19321 
19600 


2515456 
2571353 
2628072 
2685619 
2744000 


11 
11 
11 
11 
11 


6619038 
7046999 
7473401 
7898261 
8321596 


5 
5 
5 
5 
5 


.1425632 
.1551367 
1676493 
1801015 
1924941 




.007352941 
.007299270 
.007246377 
.007194245 
•007142857 


141 
142 
143 
144 
145 


19881 
20164 
20449 
20736 
21025 


2803221 
2863288 
2924207 
2985984 
3048625 


11 
11 
11 
12 
12 


8743421 
9163753 
9582607 
0000000 
0415946 


5 
5 
5 
5 
5 


2048279 
2171034 
2293215 
2414828 
2535879 




.007092199 
.007042254 
.006993007 
.006944444 
.006896552 


146 
147 
; 148 
149 
150 


21316 
21609 
21904 
22201 
22500 


3112136 
3176523 
3241792 
3307949 
3375000 


12 
12 
12 
12 
12 


0830460 
1243557 
1655251 
2065556 
2474487 


5 
5 
5 
5 
5 


2656374 
2776321 
2895725 
3014592 
3132928 




.006849315 
.006802721 
.006756757 
.006711409 
.006666667 


151 
152 
153 
154 
155 


22801 
23104 
23409 
23716 
24025 


3442951 
3511808 
3581577 
3652264 
3723875 


12 
12 
12 
12 
12 


2882057 
3288280 
3693169 
4096736 
4498996 


5 
5 
5 
5 
5 


3250740 
3368033 
3484812 
3601084 
3716854 




.006622517 
.006578947 
.006535948 
006493506 
006451613 


156 
157 
158 
159 
160 


24336 
24649 
24964 
25281 
25600 


3796416 
3869893 
3944312 
4019679 
4096000 


12 
12 
12 
12 
12 


4899960 
5299641 
5698051 
6095202 
6491106 


5 
5 
5 
5 
5 


3832126 
3946907 
4061202 
4175015 
4288352 




006410256 
006369427 
006329114 
006289308 
006250000 . 


161 
162 
163 
164 
165 


25921 
26244 
26569 
26896 
27225 


4173281 
4251528 
4330747 
4410944 
4492125 


12 
12 
12 
12 
12 


6885775 
7279221 
7671453 
8062485 
8452326 


5 
5 
5 
5 
5 


4401218 
4513618 
4625556 
4737037 
4848066 




006211180 
006172840 
006134969 
006097561 
006060606 


166 
167 
168 
169 
170 


27556 
27889 
28224 
28561 
28900 


4574296 
4657463 
4741632 
4826809 
4913000 


12 
12 
12 
13 
13 


8840987 
9228480 
9614814 
0000000 
0384048 


5 
5 
5 
5 
5 


4958647 
5068784 
5178484 
5287748 
5396583 




006024096 
005988024 
005952381 
005917160 
005882353 


171 
172 
173 
174 
175 


29241 
29584 
29929 
30276 
30625 


5000211 
5088448 
5177717 
5268024 
5359375 


13 
13 
13 
13 
13 


0766968 
1148770 
1529464 
1909060 
2287566 


5 
•5 
5 
5 
5 


5504991 
5612978 
5720546 
5827702 
5934447 




005847953 
005813953 
005780347 
005747126 
005714286 


176 
177 
178 
179 
180 


30976 
31329 
31684 
32041 
32400 


5451776 
5545233 
5639752 
5735339 
5832000 


13 
13 
13 
13 
13 


2664992 
3041347 
3416641 
3790882 
4164079 


5 
5 
5. 
5. 
5. 


6040787 
6146724 
6252263 
6357408 
6462162 




005681818 
005649718 
005617978 
005586592 
005555556 



748 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


181 
182 
183 
184 
185 


32761 
33124 
33489 
33856 
34225 


5929741 
6028568 
6128487 
6229504 
6331625 


13.4536240 
13.4907376 
13.5277493 
13.5646600 
13-6014705 


5.6566528 
5.6670511 
5.6774114 
5.6877340 
5-6980192 


•005524862 
•005494505 
•005464481 
•005434783 
.005405405 


- 186 
187 
188 
189 
190 


34596 
34969 
35344 
35721 
36100 


6434856 
6539203 
6644672 
6751269 
6859000 


13.6381817 
13.6747943 
13.7113092 
13.7477271 
13-7840488 


5.7082675 
5.7184791 
5.7286543 
5.7387936 
5-7488971 


•005376344 
.005347594 
•005319149 
•005291005 
.005263158 


- 191 
192 
193 
194 
195 


36481 
36864 
37249 
37636 
38025 


6967871 
7077888 
7189057 
7301384 
7414875 


13.8202750 
13.8564065 
13.8924440 
13.9283883 
13-9642400 


5-7589652 
5-7689982 
5.7789966 
5.7889604 
5.7988900 


.005235602 
•005208333 
•005181347 
•005154639 
.005128205 


- 196 
197 
198 
199 
300 


38416 
38809 
39204 
39601 
40000 


7529536 
7645373 
7762392 
7880599 
8000000 


14.0000000 
14.0356688 
14.0712473 
14.1067360 
14-1421356 


5-8087857 
5.8186479 
5.8284767 
5.8382725 
5.8480355 


.005102041 
•005076142 
•005050505 
•005025126 
.005000000 


"^ 201 
202 
1 203 
1 204 
• 205 


40401 
40804 
41209 
41616 
42025 


8120601 
8242408 
8365427 
8489664 
8615125 


14.1774469 
14.2126704 
14.2478068 
14.2828569 
14.3178211 


5-8577660 
5-8674643 
5-8771307 
5-8867653 
5-8963685 


•004975124 
•004950495 
•004926108 
•004901961 
.004878049 


-1 206 

207 

, 208 

\ 209 

310 


42436 
42849 
43264 
43681 
44100 


8741816 
8869743 
8998912 
9129329 
9261000 


14.3527001 
14.3874946 
14.4222051 
14.4568323 
14.4913767 


5-9059406 
5.9154817 
5.9249921 
5-9344721 
5-9439220 


•004854369 
•004830918 
•004807692 
•004784689 
.004761905 


211 

212 

1 213 

t 214 

215 


44521 
44944 
45369 
45796 
46225 


9393931 
9528128 
9663597 
9800344 
9938375 


14.5258390 
14.5602198 
14.5945195 
14.6287388 
14.6628783 


5-9533418 
5-9627320 
5-9720926 
5.9814240 
5.9907264 


.004739336 
.004716981 
.004694836 
.004672897 
.004651163 


^ 216 
217 
1 218 
1 219 
( 220 


46656 
47089 
47524 
47961 
48400 


10077696 
10218313 
10360232 
10503459 
10648000 


14.6969385 
14.7309199 
14.7648231 
14.7986486 
14.8323970 


fe. 0000000 
6.0092450 
6.0184617 
6.0276502 
B0368107 


•004629630 
•004608295 
•004587156 
.004566210 
.004545455 


- 221 
222 
223 

, 224 
225 


48841 
49284 
49729 
50176 
50625 


10793861 
10941048 
11089567 
11239424 
11390625 


14.8660687 
14.8996644 
14.9331845 
14.9666295 
15.0000000 


6.0459435 
6.0550489 
6-0641270 
6-0731779 
6-0822020 


•004524887 
•004504505 
•004484305 
•004464286 
•004444444 


1 226 
227 

' 228 
229 
230 


51076 
51529 
51984 
52441 
52900 


11543176 
11697083 
11852352 
12008989 
12167000 


15.0332964 
15.0665192 
15.0096689 
15.1327460 
15.1657509 


6-0911994 
6.1001702 
6-1091147 
6.1180332 
6.1269257 


•004424779 
•004405286 
•004385965 
•004366812 
•004347826 


231 
232 
233 
234 
235 


53361 
53824 
54289 
54756 
55225 


12326391 
12487168 
12649337 
12812904 
12977875 


15.1986842 
15.2315462 
15.2643375 
15.2970585 
15.3297097 


6-1357924 
6.1446337 
6.1534495 
6.1622401 
6.1710058 


•004329004 
•004310345 
•004291845 
•004273504 
.004255319 


236 
237 
, 238 
239 
240 


55696 
56169 
56644 
57121 
57600 


13144256 
13312053 
13481272 
13651919 
13824000 


15.3622915 
15.3948043 
15.4272486 
15.4596248 
15.4919334 


6.1797466 
6.1884628 
6.1971544 
6.2058218 
6.2144650 


•004237288 
•004219409 
•004201681 
•004184100 
.004166667 








749 









TABLE XXXIL— SQUARES, CUBES, 


SQUARE ROOTS, 


No. 


Squares, 


Cubeso 


Square Roots, 


Cube Roots. 


Reciprocals. 


241 


58081 


13997521 


15.5241747 


6.2230843 


.004149378 


242 


58564 


14172488 


15.5563492 


6 


2316797 


.004132231 


243 


59049 


14348907 


15.5884573 


6 


2402515 


.004115226 


244 


59536 


14526784 


15.6204994 


6 


2487998 


.004098361 


245 


60025 


14706125 


15.6524758 


6 


2573248 


.004081633 


246 


60516 


14886936 


15.6843871 


6 


2658266 


.004065041 


247 


61009 


15069223 


15.7162336 


6 


2743054 


.004048583 


248 


61504 


15252992 


15.7480157 


6 


2827613 


.004032258 


249 


62001 


15438249 


15.7797338 


6 


2911946 


.004016064 


350 


62500 


15625000 


15.8113883 


6 


2996053 


.004000000 


251 


63001 


15813251 


15.8429795 


6 


3079935 


.003984064 


252 


63504 


16003008 


15.8745079 


6 


3163596 


.003968254 


253 


64009 


16194277 


15.9059737 


6 


3247035 


.003952569 


254 


64516 


16387064 


15.9373775 


6 


3330256 


.003937008 


255 


65025 


16581375 


15.9687194 


6 


3413257 


.003921569 


256 


65536 


16777216 


16.0000000 


6 


3496042 


.003906250 


257 


66049 


16974593 


16.0312195 


6 


3578611 


.003891051 


258 


66564 


17173512 


16.0623784 


6 


3660968 


.003875969 


259 


67081 


17373979 


16.0934769 


6 


3743111 


.003861004 


260 


67600 


17576000 


16.1245155 


6 


3825043 


.003846154 


261 


68121 


17779581 


16.1554944 


6 


3906765 


.003831418 


262 


68644 


17984728 


16.1864141 


6 


3988279 


.003816794 


263 


69169 


18191447 


16.2172747 


6 


4069585 


.003802281 


264 


69696 


18399744 


16.2480768 


6 


4150687 


.003787879 


265 


70225 


18609625 


16.2788206 


6 


4231583 


.003773585 


266 


70756 


18821096 


16.3095064 


6 


4312276 


.003759398 


267 


71289 


19034163 


16-3401346 


6 


4392767 


.003745318 


268 


71824 


19248832 


16.3707055 


6 


4473057 


.003731343 


269 


72361 


19465109 


16.4012195 


6 


4553148 


»003717472 


270 


72900 


19683000 


16.4316767 


6 


4633041 


.003703704 


271 


73441 


19902511 


16.4620776 


6 


4712736 


.003690037 


272 


73984 


20123648 


16.4924225 


6 


4792236 


.003676471 


273 


74529 


20346417 


16.5227116 


6 


4871541 


.003663004 


274 


75076 


20570824 


16.5529454 


6 


4950653 


.003649635 


275 


75625 


20796875 


16.5831240 


6 


5029572 


.003636364 


276 


76176 


21024576 


16.6132477 


6 


5108300 


.003623188 


277 


76729 


21253933 


16.6433170 


6 


5186839 


.003610108 


278 


77284 


21484952 


16.6733320 


6 


5265189 


.003597122 


279 


77841 


21717639 


16.7032931 


6 


5343351 


.003584229 


280 


78400 


21952000 


16.7332005 


6 


5421326 


-003571429 


281 


78961 


22188041 


16.7630546 


6 


.5499116 


.003558719 


282 


79524 


22425768 


16-7928556 


6 


.5576722 


.003546099 


283 


80089 


22665187 


16.8226038 


6 


.5654144 


.003533569 


284 


80656 


22906304 


16-8522995 


6 


.5731385 


.003521127 


285 


81225 


23149125 


16-8819430 


6 


5808443 


-003508772 


286 


81796 


23393656 


16-9115345 


6 


5885323 


.003496503 


287 


82369 


23639903 


16.9410743 


6 


.5962023 


.003484321 


288 


82944 


23887872 


16.9705627 


6 


.6038545 


.003472222 


289 


83521 


24137569 


17-0000000 


6 


6114890 


.003460208 


290 


84100 


24389000 


17.0293864 


6 


6191060 


-003448276 


291 


84681 


24642171 


17-0587221 


6 


6267054 


.003436426 


292 


85264 


24897088 


17-0880075 


6 


6342874 


.003424658 


293 


85849 


25153757 


17-1172428 


6 


6418522 


.003412969 


294 


86436 


25412184 


17-1464282 


6 


6493998 


.003401361 


295 


87025 


25672375 


17-1755640 


6 


6569302 


.003389831 


296 


87616 


25934336 


17.2046505 


6 


6644437 


.003378378 


297 


88209 


26198073 


17.2336879 


6 


6719403 


.003367003 


298 


88804 


26463592 


17-2626765 


6 


6794200 


.003355705 


299 


89401 


26730899 


17-2916165 


6 


6868831 


.003344482 


300 


90000 


27000000 


17.3205081 


6.6943295 


.003333333 



750 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes, 


Square Roots. 


Cube Roots. 


Reciprocals, 


301 
302 
303 
304 
305 


90601 
91204 
91809 
92416 
93025 


27270901 
27543608 
27818127 
28094464 
28372625 


17.3493516 
17.3781472 
17.4068952 
17.4355958 
17.4642492 


6.7017593 
6.7091729 
6.7165700 
6.7239508 
6.7313155 


.003322259 
.003311258 
.003300330 
.003289474 
.003278689 


306 
307 
308 
309 
310 


93636 
94249 
94864 
95481 
96100 


28652616 
28934443 
29218112 
29503629 
29791000 


17.4928557 
17.5214155 
17.5499288 
17.5783958 
17.6068169 


6.7386641 
6.7459967 
6.7533134 
6.7606143 
6.7678995 


.003267974 
.003257329 
•003246753 
.003236246 
.003225806 


311 
312 
313 
314 
315 


96721 
97344 
97969 
98596 
99225 


30080231 
30371328 
30664297 
30959144 
31255875 


17.6351921 
17.6635217 
17.6918060 
17.7200451 
17.7482393 


6.7751690 
6.7824229 
6.7896613 
6.7968844 
6.8040921 


.003215434 
.003205128 
.003194888 
.003184713 
.003174603 


316 
317 
318 
319 
330 


99856 
100489 
101124 
101761 
102400 


31554496 
31855013 
32157432 
32461759 
32768000 


17.7763888 
17.8044938 
17.8325545 
17.8605711 
17.8885438 


6.8112847 
6.8184620 
6.8256242 
6.8327714 
6.8399037 


.003164557 
.003154574 
.003144654 
.003134796 
.003125000 


321 
322 
323 
324 
325 


103041 
103684 
104329 
104976 
105625 


33076161 
33386248 
33698267 
34012224 
34328125 


17.9164729 
17.9443584 
17.9722008 
18.0000000 
18.0277564 


6.8470213 
6.8541240 
6.8612120 
6.8682855 
6.8753443 


.003115265 
.003105590 
.003095975 
.003086420 
.003076923 


326 
327 
328 
829 
330 


106276 
106929 
107584 
108241 
108900 


34645976 
34965783 
35287552 
35611289 
35937000 


18.0554701 
18.0831413 
18.1107703 
18.1383571 
18.1659021 


6.8823888 
6.8894188 
6.8964345 
6.9034359 
6.9104232 


.003067485 
.003058104 
.003048780 
.003039514 
.003030303 


331 
332 
333 
334 
335 


109561 
110224 
110889 
111556 
112225 


36264691 
36594368 
36926037 
37259704 
37595375 


18.1934054 
18.2208672 
18.2482876 
18.2756669 
18.3030052 


6-9173964 
6.9243556 
6.9313008 
6.9382321 
6.9451496 


.003021148 
.003012048 
•003003003 
.002994012 
.002985075 


336 
337 
338 
339 
340 


112896 
113569 
114244 
114921 
115600 


37933056 
38272753 
38614472 
38958219 
39304000 


18.3303028 
18.3575598 
18.3847763 
18.4119526 
18.4390889 


6.9520533 
6.9589434 
6.9658198 
6.9726826 
6.9795321 


.002976190 
.002967359 
.002958580 
•002949853 
.002941176 


341 
342 
343 
344 
345 


116281 
116964 
117649 
118336 
119025 


39651821 
40001688 
40353607 
40707534 
41063625 


18.4661853 
18.4932420 
18.5202592 
18.5472370 
18.5741756 


6.9863681 
6.9931906 
7.0000000 
7-0067962 
7.0135791 


.002932551 
.002923977 
.002915452 
.002906977 
.002898551 


346 
347 
348 
349 
350 


119716 
120409 
121104 
121801 
122500 


41421736 
41781923 
42144192 
42508549 
42875000 


18.6010752 
18.6279360 
18.6547581 
18.6815417 
18.7082869 


7.0203490 
7.0271058 
7.0338497 
7.0405806 
7.0472987 


.002890173 
.002881844 
.002873563 
.002865330 
.002857143 


351 
352 
353 
354 
355 


123201 
123904 
124609 
125316 
126025 


43243551 
43614208 
43986977 
44361864 
44738875 


18.7349940 
18.7616630 
18.7882942 
18.8148877 
18.8414437 


7.0540041 
7.0606967 
7.0673767 
7.0740440 
7.0806988 


.002849003 
.002840909 
.002832861 
.002824859 
.002816901 


356 
357 
358 
359 
360 


126736 
127449 
128164 
128881 
129600 


45118016 
45499293 
45882712 
46268279 
46656000 


18.8679623 
18.8944436 
18.9208879 
18.9472953 
18.9736660 


7.0873411 
7.0939709 
7.1005885 
7.1071937 
7.1137866 


.002808989 
.002801120 
.002793296 
.002785515 
.002777778 



751 



TABLE XXXII.— SQUARES, CUBES, SQUARE ROOTS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


361 
362 
363 
364 
365 


130321 
131044 
131769 
132496 
133225 


47045881 
47437928 
47832147 
48228544 
48627125 


19.0000000 
19.0262976 
19.0525589 
19.0787840 
19.1049732 


7.1203674 
7.1269360 
7.1334925 
7.1400370 
7.1465695 


.002770083 
.002762431 
.002754821 
.002747253 
.002739726 


366 
367 
368 
369 
370 


133956 
134689 
135424 
136161 
136900 


49027896 
49430863 
49836032 
50243409 
50653000 


19.1311265 
19.1572441 
19.1833261 
19.2093727 
19.2353841 


7.1530901 
7.1595988 
7.1660957 
7.1725809 
7.1790544 


.002732240 
.002724796 
.002717391 
.002710027 
.002702703 


371 
372 
373 
374 
375 


137641 
138384 
139129 
139876 
140625 


51064811 
51478848 
51895117 
52313624 
52734375 


19.2613603 
19.2873015 
19.3132079 
19.3390796 
19.3649167 


7.1855162 
7.1919663 
7.1984050 
7.2048322 
7.2112479 


.002695418 
.002688172 
.002680965 
.002673797 
.002666667 


376 
377 
378 
379 
380 


141376 
142129 
142884 
143641 
144400 


53157376 
53582633 
54010152 
54439939 
54872000 


19.3907194 
19.4164878 
19.4422221 
19.4679223 
19.4935887 


7.2176522 
7.2240450 
7.2304268 
7.2367972 
7.2431565 


.002659574 
.002652520 
.002645503 
.002638522 
.002631579 


381 
382 
383 
384 
, 385 . 


145161 
145924 
146689 
147456 
148225 


55306341 
55742968 
56181887 
56623104 
57066625 


19.5192213 
19.5448203 
19.5703858 
19.5959179 
19.6214169 


7.2495045 
7.2558415 
7.2621675 
7.2684824 
7.2747864 


.002624672 
.002617801 
.002610966 
.002604167 
.002597403 


386 
387 
388 
389 
390 


148996 
149769 
150544 
151321 
152100 


57512456 
57960603 
58411072 
58863869 
59319000 


19.6468827 
19.6723156 
19.6977156 
19.7230829 
19.7484177 


7.2810794 
7.2873617 
7.2936330 
7.2998936 
7.3061436 


.002590674 
.002583979 
.002577320 
.002570694 
.002564103 


391 
392 
393 
394 
395 


152881 
153664 
154449 
155236 
156025 


59776471 
60236288 
60698457 
61162984 
61629875 


19.7737199 
19.7989899 
19.8242276 
19.8494332 
19.8746069 


7.3123828 
7.3186114 
7.3248295 
7.3310369 
7.3372339 


.002557545 
.002551020 
.002544529 
.002538071 
.002531646 


396 
397 
398 
399 
400 


156816 
157609 
158404 
159201 
160000 


62099136 
62570773 
63044792 
63521199 
64000000 


19.8997487 
19.9248588 
19.9499373 
19.9749844 
20.0000000 


7.3434205 
7.3495966 
7.3557624 
7.3619178 
7.3680630 


.002525253 
.002518892 
.002512563 
.002506266 
.002500000 _ 


401 
402 
403 
404 
405 


160801 
161604 
162409 
163216 
164025 


644B1201 
64964808 
65450827 
65939264 
66430125 


20.0249844 
20.0499377 
20.0748599 
20.0997512 
20.1246118 


7.3741979 
7.3803227 
7.3864373 
7.3925418 
7.3986363 


.002493766 
.002487562 
.002481390 
.002475248 
.002469136 


406 
407 
408 
409 
410 


164836 
165649 
166464 
167281 
168100 


66923416 
67419143 
67917312 
68417929 
68921000 


20.1494417 
20.1742410 
20.1990099 
20.2237484 
20.248.4567 


7.4047206 
7.4107950 
7.4168595 
7.4229142 
7.4289589 


.002463054 
.002457002 
.002450980 
.002444988 
.002439024 


411 
412 
413 
414 
415 


168921 
169744 
170569 
171396 
172225 


69426531 
69934528 
70444997 
70957944 
71473375 


20.2731349 
20.2977831 
20.3224014 
20.3469899 
20.3715488 


7.4349938 
7.4410189 
7.4470342 
7.4530399 
7.4590359 


.002433090 
.002427184 
.002421308 
.002415459 
.002409639 


416 
417 
418 
419 
430 


173056 
173889 
174724 
175561 
176400 


71991296 
72511713 
73034632 
73560059 
74088000 


20.3960781 
20.4205779 
20.4450483 
20.4694895 
20-4939015 


7.4650223 
7.4709991 
7.4769664 
7.4829242 
7.4888724 


.002403846 
.002398082 
.002392344 
.002386635 
.00238C^5?t 



752 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


121 
422 
423 
424 
425 


177241 
178084 
178929 
179776 
180625 


74618461 
75151448 
75686967 
76225024 
76765625 


20 
20 
20 
20 
20 


.5182845 
.5426386 
.5669638 
5912603 
6155281 


7.4948113 
7.5007406 
7.5066607 
7.5125715 
7.5184730 


.002375297 
.002369668 
.002364066 
.002358491 
-002352941 


426 
427 
428 
429 
430 


181476 
182329 
183184 
184041 
184900 


77308776 
77854483 
78402752 
78953589 
79507000 


20 
20 
20 
20 
20 


6397674 
6639783 
6881609 
7123152 
7364414 


7.5243652 
7.5302482 
7.5361221 
7.5419867 
7.5478423 


.002347418 
.002341920 
.002336449 
.002331002 
.002325581 


431 
432 
433 
434 
435 


185761 
186624 
187489 
188356 
189225 


80062991 
80621568 
81182737 
81746504 
82312875 


20 
20 
20 
20 
20 


7605395 
7846097 
8086520 
8326667 
8566536 


7.5536888 
7.5595263 
7.5653548 
7.5711743 
7.5769849 


.002320186 
.002314815 
.002309469 
.002304147 
.002298851 


436 
437 
438 
439 
440 


190096 
190969 
191844 
192721 
193600 


82881856 
83453453 
84027672 
84604519 
85184000 


20 
20 
20 
20 
„ 20 


.8806130 
.9045450 
.9284495 
9523268 
9761770 


7.5827865 
7.5885793 
7-5943633 
7.6001385 
7-6059049 


.002293578 
.002288330 
.002283105 
.002277904 
-002272727 


441 
442 
443 
444 
445 


194481 
195364 
196249 
197136 
198025 


85766121 
86350888 
86938307 
87528384 
88121125 


21 
21 
21 
21 
21 


0000000 
0237960 
0475652 
0713075 
0950231 


7-6116626 
7-6174116 
7-6231519 
7-6288837 
7-6346067 


.002267574 
.002262443 
.002257336 
.002252252 
.002247191 


446 
447 
448 
449 
450 


198916 
199809 
200704 
201601 
202500 


88716536 
89314623 
89915392 
90518849 
91125000 


21 
21 
21 
21 
21 


1187121 
1423745 
1660105 
1896201 
2132034 


7-6403213 
7-6460272 
7.6517247 
7.6574138 
7-6630943 


.002242152 
.002237136 
.002232143 
.002227171 
.002222222 


451 
452 
453 
454 
455 


203401 
204304 
205209 
206116 
207025 


91733851 
92345408 
92959677 
93576664 
94196375 


21 
21 
21 
21 
21 


2367606 
2602916 
2837967 
3072758 
3307290 


7-6687665 
7-6744303 
7-6800857 
7-6857328 
7-6913717 


.002217295 
.002212389 
.002207506 
•002202643 
.002197802 


456 
457 
458 
459 
. 460 


207936 
208849 
209764 
210681 
211600 


94818816 
95443993 
96071912 
96702579 
97336000 


21 
21 
21 
21 
21 


3541565 
3775583 
4009346 
4242853 
4476106 


7.6970023 
7-7026246 
7.7082388 
7.7138448 
7-7194426 


.002192982 
.002188184 
.002183406 
.002178649 
.002173913 


461 
462 
463 
464 
465 


212521 
213444 
214369 
215296 
216225 


97972181 
98611128 
99252847 
99897344 
100544625 


21 
21 
21 
21 
21 


4709106 
4941853 
5174348 
5406592 
5638587 


7.7250325 
7.7306141 
7.7361877 
7.7417532 
7-7473109 


.002169197 
.002164502 
.002159827 
.002155172 
.002150538 


466 
467 
468 
469 
470 


217156 
218089 
219024 
219961 
220900 


101194696 
101847563 
102503232 
103161709 
103823000 


21 
21 
21 
21 
21 


5870331 
6101828 
6333077 
6564078 
6794834 


7-7528606 
7-7584023 
7.7639361 
7.7694620 
7-7749801 


.002145923 
.002141328 
.002136752 
.002132196 
.002127660 


471 
472 
473 
474 
475 


221841 
222784 
223729 
224676 
225625 


104487111 
105154048 
105823817 
106496424 
107171875 


21. 
21 
21 
21 
21 


7025344 
7255610 
7485632 
7715411 
7944947 


7.7804904 
7.7859928 
7.7914875 
7.7969745 
7.8024538 


.002123142 
.002118644 
.002114165 
.002109705 
.002105263 


476 
477 
478 
479 
480 


226576 
227529 
228484 
229441 
230400 


107850176 
108531333 
109215352 
109902239 
110592000 


21 
21 
21 
21 
21 


8174242 
8403297 
8632111 
8860686 
9089023 


7.8079254 
7.8133892 
7.8188456 
7.8242942 
7.8297353 


.002100840 
.002096436 
.002092050 
.002087683 
.002083333 



753 



TABLE XXXII.— SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


481 
482 
483 
484 
. 485 


231361 
232324 
233289 
234256 
235225 


111284641 
111980168 
112678587 
113379904 
114084125 


21.9317122 
21.9544984 
21.9772610 
22.0000000 
22.0227155 


7.8351688 
7.8405949 
7.8460134 
7.8514244 
7.8568281 


.002079002 
.002074689 
.002070393 
.002066116 
.002061856 


486 
487 
488 
489 
490 


236196 
237169 
238144 
239121 
240100 


114791256 
115501303 
116214272 
116930169 
117649000 


22.0454077 
22.0680765 
22.0907220 
22.1133444 
22.1359436 


7.8622242 
7.8676130 
7.8729944 
7.8783684 
7.8837352 


.002057613 
.002053388 
.002049180 
.002044990 
.002040816 


491 
492 
493 
494 
495 


241081 
242064 
243049 
244036 
245025 


118370771 
119095488 
119823157 
120553784 
121287375 


22.1585198 
22.1810730 
22.2036033 
22.2261108 
22.2485955 


7.8890946 
7.8944468 
7.8997917 
7.9051294 
7.9104599 


.002036660 
.002032520 
.002028398 
.002024291 
.002020202 


496 
497 
498 
499 
500 


246016 
247009 
248004 
249001 
250000 


122023936 
122763473 
123505992 
124251499 
125000000 


22-2710575 
22.2934968 
22.3159136 
22.3383079 
22.3606798 


7.9157832 
7.9210994 
7.9264085 
7.9317104 
7.9370053 


.002016129 
.002012072 
.002008032 
.002004008 
.002000000 


501 
502 
503 
504 
505 


251001 
252004 
253009 
254016 
255025 


125751501 
126506008 
127263527 
128024064 
128787625 


22.3830293 
22.4053565 
22.4276615 
22.4499443 
22.4722051 


7.9422931 
7.9475739 
7.9528477 
7.9581144 
7.9633743 


.001996008 
.001992032 
.001988072 
.001984127 
.001980198 


506 
507 
508 
509 
510 


256036 
257049 
258064 
259081 
260100 


129554216 
130323843 
131096512 
131872229 
132651000 


22.4944438 
22.5166605 
22.5388553 
22.5610283 
22.5831796 


7.9686271 
7.9738731 
7.9791122 
7.9843444 
7.9895697 


.001976285 
.001972387 
.001968504 
.001964637 
.001960784 


511 
512 
513 
514 
515 


261121 
262144 
263169 
264196 
265225 


133432831 
134217728 
135005697 
135796744 
136590875 


22.6053091 
22.6274170 
22.6495033 
22.6715681 
22.6936114 


7.9947883 
8.0000000 
8.0052049 
8.0104032 
8.0155946 


.001956947 
.001953125 
.001949318 
.001945525 
.001941748 


516 
517 
518 
519 
520 


266256 
267289 
268324 
269361 
270400 


137388096 
138188413 
138991832 
139798359 
140608000 


22.7156334 
22.7376340 
22.7596134 
22.7815715 
22.8035085 


8.0207794 
8.0259574 
8.0311287 
8.0362935 
8.0414515 


.001937984 
.001934236 
.001930502 
.001926782 
.001923077 , 


521 
522 
523 
524 
525 


271441 
272484 
273529 
274576 
275625 


141420761 
142236648 
143055667 
143877824 
144703125 


22.8254244 
22.8473193 
22.8691933 
22.8910463 
22.9128785 


8.0466030 
8.0517479 
8.0568862 
8.0620180 
8.0671432 


.001919386 
.001915709 
.001912046 
.001908397 
.001904762 


526 
527 
528 
529 
530 


276676 
277729 
278784 
279841 
280900 


145531576 
146363183 
147197952 
148035889 
148877000 


22.9346899 
22.9564806 
22.9782506 
23.0000000 
23.0217289 


8.0722620 
8.0773743 
8.0824800 
8.0875794 
8.0926723 


.001901141 
.001897533 
.001893939 
.001890359 
.001886792 


531 
532 
533 
534 
535 


281961 
283024 
284089 
285156 
286225 


149721291 
150568768 
151419437 
152273304 
153130375 


23.0434372 
23.0651252 
23.0867928 
23.1084400 
23.1300670 


8.0977589 
8.1028390 
8.1079128 
8.1129803 
8.1180414 


.001883239 
.001879699 
.001876173 
.001872659 
.001869159 


536 
537 
538 
539 
540 


287296 
288369 
289444 
290521 
291600 


153990656 
154854153 
155720872 
156590819 
157464000 


23.1516738 
23.1732605 
23.1948270 
23.2163735 
23.2379001 


8.1230962 
8.1281447 
8.1331870 
8.1382230 
8.1432529 


.001865672 
.001862197 
.0018587ii6 
.001855288 
.001851852 



754 



CUBE ROOTS. AND RECIPROCALS. 



No. 


Squares. 


Cubes, 


Square Roots. 


Cube Roots. 


Reciprocals. 


541 
542 
543 
544 
545 


292681 
293764 
294849 
295936 
297025 


158340421 
159220088 
160103007 
160989184 
161878625 


23.2594067 
23.2808935 
23.3023604 
23.3238076 
23.3452351 


8.1482765 
8.1532939 
8.1583051 
8.1633102 
8.1683092 


.001848429 
.001845018 
.001841621 
.001838235 
.001834862 


546 
547 
548 
549 
550 


298116 
299209 
300304 
301401 
302500 


162771336 
163667323 
164566592 
165469149 
166375000 


23.3666429 
23.3880311 
23.4093998 
23.4307490 
23.4520788 


8.1733020 
8.1782888 
8.1832695 
8.1882441 
8.1932127 


.001831502 
.001828154 
.001824818 
.001821494 
.001818182 


551 
552 
553 
554 
555 


303601 
304704 
305809 
306916 
308025 


167284151 
168196608 
169112377 
170031464 
170953875 


23.4733892 
23.4946802 
23.5159520 
23.5372046 
23.5584380 


8.1981753 
8.2031319 
8.2080825 
8.2130271 
8.2179657 


.001814882 
.001811594 
.001808318 
.001805054 
.001801802 


556 
557 
558 
559 
560 


309136 
310249 
311364 
312481 
313600 


171879616 
172808693 
173741112 
174676879 
175616000 


23.5796522 
23-6008474 
23.6220236 
23.6431808 
23.6643191 


8^2228985 
8.2278254 
8.2327463 
8-2376614 
8.2425706 


.001798561 
.001795332 
.001792115 
.001788909 
.001785714 


561 
562 
563 
564 
565 


314721 
315844 
316969 
318096 
319225 


176558481 
177504328 
178453547 
179406144 
180362125 


23.6854386 
23.7065392 
23.7276210 
23.7486842 
23.7697286 


8.2474740 
8.2523715 
8.2572633 
8.2621492 
8.2670294 


.001782531 
.001779359 
.001776199 
.001773050 
.001769912 


566 
567 
568 
569 
570 


320356 
321489 
322624 
323761 
324900 


181321496 
182284263 
183250432 
184220009 
185193000 


23.7907545 
23.8117618 
23.8327506 
23.8537209 
23.8746728 


8.2719039 
8.2767726 
8.2816355 
8.2864928 
8.2913444 


.001766784 
.001763668 
.001760563 
=001757469 
.001754386 


571 
572 
573 
574 
575 


326041 
327184 
328329 
329476 
330625 


186169411- 

187149248 

188132517 

189119224 

190109375 


23.8956063 
23.9165215 
23.9874184 
23.9582971 
23.9791576 


8.2961903 
8.3010304 
8.3058651 
8.3106941 
8=3155175 


.001751313 
.001748252 
.001745201 
.001742160 
.001739130 


576 
577 
578 
579 
580 


331776 
332929 
334084 
335241 
336400 


191102976 
192100033 
193100552 
194104539 
195112000 


24.0000000 
24.0208243 
24.0416306 
24.0624188 
24.0831891 


8.3203353 
8.3251475 
8.3299542 
8.3347553 
8-3395509 


.001736111 
.001733102 
.001730104 
.001727116 
.001724138 


581 
582 
583 
584 
585 


337561 
338724 
339889 
341056 
342225 


196122941 
197137368 
198155287 
199176704 
200201625 


24.1039416 
24.1246762 
24.1453929 
24.1660919 
24.1867732 


8.3443410 
8.3491256 
8.3539047 
8.35867^4 
8.3634466 


.001721170 
.001718213 
.001715266 
.001712329 
.001709402 


586 
587 
588 
589 
590 


343396 
344569 
345744 
346921 
348100 


201230056 
202262003 
203297472 
204336469 
205379000 


24.2074369 
24.2280829 
24.2487113 
24.2693222 
24.2899156 


8.3682095 
8.3729668 
8.3777188 
8.3824653 
8.3872065 


.001706485 
.001703578 
.001700680 
.001697793 
.001694915 


591 
592 
593 
594 
595 


349281 
350464 
351649 
352836 
354025 


206425071 
207474688 
208527857 
209584584 
210644875 


24.3104916 
24.3310501 
24 3515913 
24.3721152 
24.3926218 


8.3919423 
8.3966729 
8.4013981 
8.4061180 
8.4108326 


.001692047 
.001689189 
.001686341 
.001683502 
.001680672 


596 
597 
598 
599 
600 


355216 
356409 
357604 
358801 
360000 


211708736 
212776173 
213847192 
214921799 
216000000 


24.4131112 
24.4335834 
24.4540385 
24.4744765 
24.4948974 


8.4155419 
8.4202460 
8.4249448 
8.4296383 
8.4343267 


.001677852 
.001675042 
.001672241 
.001669449 
.001666667 



755 



TABLE XXXII.— SQUARES. CUBES. SQUARE ROOTS. 



No. 


Squares, 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals, 


601 
602 
603 
604 
605 


361201 
362404 
363609 
364816 
366025 


217081801 
218167208 
219256227 
220348864 
221445125 


24.5153013 
24.5356883 
24.5560583 
24.5764115 
24.5967478 


8.4390098 
8.4436877 
8.4483605 
8.4530281 
8.4576906 


.001663894 
.001661130 
.001658375 
.001655629 
.001652893 _ 


606 
607 
608 
609 
610 


367236 
368449 
369664 
370881 
372100 


222545016 
223648543 
224755712 
225866529 
226981000 


24.6170673 
24.6373700 
24.6576560 
24.6779254 
24.6981781 


8.4623479 
8.4670001 
8.4716471 
8.4762892 
8.4809261 


.001650165 
.001647446 
.001644737 
.001642036 
.001639344 


611 
612 
613 
614 
615 


373321 
374544 
375769 
376996 
378225 


228099131 
229220928 
230346397 
231475544 
232608375 


24.7184142 
24.7386338 
24.7588368 
24.7790234 
24.7991935 


8.4855579 
8.4901848 
8.4948065 
8.4994233 
8.5040350 


.001636661 
.001633987 
.001631321 
.001628664 
.001626016 


616 
617 
618 
619 
630 


379456 
380689 
381924 
383161 
384400 


233744896 
234885113 
236029032 
237176659 
238328000 


24.8193473 
24.8394847 
24.8596058 
24.8797106 
24.8997992 


8.5086417 
8.5132435 
8.5178403 
8.5224321 
8.5270189 


.001623377 
.001620746 
.001618123 
.001615509 
.001612903 


621 

622 

623 

, 624 

. 625 


385641 
386884 
388129 
389376 
390625 


239483061 
240641848 
241804367 
242970624 
244140625 


24.9198716 
24.9399278 
24.9599679 
24.9799920 
25.0000000 


8.5316009 
8.5361780 
8.5407501 
8.5453173 
8.5498797 


.001610306 
.001607717 
.001605136 
.001602564 
.001600000 


626 
627 
628 
629 
630 


391876 
393129 
394384 
395641 
396900 


245314376 
246491883 
247673152 
248858189 
250047000 


25.0199920 
25.0399681 
25.0599282 
25.0798724 
25.0998008 


8.5544372 
8.5589899 
8.5635377 
8.5680807 
8.5726189 


.001597444 
.001594896 ^ 
.001592357 
.001589825 
.001587302 


631 
632 
633 
634 
635 


398161 
399424 
400689 
401956 
403225 


251239591 
252435968 
253636137 
254840104 
256047875 


25.1197134 
25.1396102 
25.1594913 
25.1793566 
25.1992063 


8.5771523 
8.5816809 
8.5862047 
8.5907238 
8.5952380 


.001584786 
.001582278 
.001579779 
.001577287 
.001574803 


636 
637 
638 
639 
640 


404496 
405769 
407044 
408321 
409600 


257259456 
258474853 
259694072 
260917119 
262144000 


25.2190404 
25.2388589 
25.2586619 
25.2784493 
25.2982213 


8.5997476 
8.6042525 
8.6087526 
8.6132480 
8.6177388 


.001572327 
.001569859 
.001567398 
.001564945 
.001562500 


641 
642 
643 
644 
645 


410881 
412164 
413449 
414736 
416025 


263374721 
264609288 
265847707 
267089984 
268336125 


25.3179778 
25.3377189 
25.3574447 
25.3771551 
25.3968502 


8.6222248 
8.6267063 
8.6311830 
8.6356551 
8.6401226 


.001560062 
.001557632 
.001555210 
.001552795 
.001550388 


646 
647 
648 
649 
650 


417316 
418609 
419904 
421201 
422500 


269586136 
270840023 
272097792 
273359449 
274625000 


25.4165301 
25.4361947 
25.4558441 
25.4754784 
25.4950976 


8.6445855 
8.6490437 
8.6534974 
8.6579465 
8.6623911 


.001547988 
.001545595 
.001543210 
.001540832 
.001538462 


651 
652 
653 
654 
_655 


423801 
425104 
426409 
427716 
429025 


275894451 
277167808 
278445077 
279726264 
281011375 


25.5147016 
25.5342907 
25.5538647 
25.5734237 
25.5929fi78 


8.6668310 
8.6712665 
8.6756974 
8.6801237 
8.6845456 


.001536098 
.001533742 
.001531394 
.001529052 
.001526718 


656 
657 
658 
659 
660 


430336 
431649 
432964 
434281 
435600 


282300416 
283593393 
284890312 
286191179 
287496000 


25.6124969 
25.6320112 
25.6515107 
25.6709953 
25.6904652 


8.6889630 
8.6933759 
8.6977843 
8.7021882 
8.7065877 


.001524390 

.001522070 

.001519757 

.00151745ir 

.001515152 



756 



CUBE ROOTS, AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


661 
662 
663 
664 
665 


436921 
438244 
439569 
440896 

442225 


288804781 
290117528 
291434247 
292754944 
294079625 


25.7099203 
25.7293607 
25.7487864 
25.7681975 
25.7875939 


8.7109827 
8.7153734 
8.7197596 
8.7241414 
8-7285187 


.001512859 ^. 

.001510574 

.001508296 

.001506024 

.001503759 


666 
667 
668 
669 
670 


443556 
444889 
446224 
447561 
448900 


295408296 
296740963 
298077632 
299418309 
300763000 


25-8069758 
25.8263431 
25.8456960 
25.8650343 
25-8843582 


8.7328918 
8.7372604 
8.7416246 
8.7459846 
8.7503401 


.001501502 
.001499250 
.001497006 
.001494768 
.001492537 , 


671 
672 
673 
674 
675 


450241 
451584 
452929 
454276 
455625 


302111711 
303464448 
304821217 
306182024 
307546875 


25.9036677 
25.9229628 
25-9422435 
25.9615100 
25-9807621 


8.7546913 
8.7590383 
8.7633809 
8.7677192 
8-7720532 


.001490313 
.001488095 
.001485884 
.001483680 
-001481481 


676 
677 
678 
679 
680 


456976 
458329 
459684 
461041 
462400 


308915776 
310288733 
311665752 
313046839 
314432000 


26-0000000 
26.0192237 
26.0384331 
26.0576284 
26-0768096 


8.7763830 
8.7807084 
8.7850296 
8.7893466 
8-7936593 


.001479290 
.001477105 
.001474926 
.001472754 
-001470588 


681 
682 
683 
684 
685 


463761 
465124 
466489 
467856 
469225 


315821241 
317214568 
318611987 
320013504 
321419125 


26.0959767 
26.1151297 
26.1342687 
26.1533937 
26-17250AV 


8.7979679 
8.8022721 
8-8065722 
8.8108681 
8.8151598 


.001468429 
.001466276 
.001464129 
.001461988 
-001459854 


686 
687 
688 
689 
690 


470596 
471969 
473344 
474721 
476100 


322828856 
324242703 
325660672 
327082769 
328509000 


26.19160:.7 
26.2106848 
26.22975 1 
26.2488095 
26-2678511 


8.8194474 
8.8237307 
8-8280099 
8-8322850 
8-8365559 


.001457726 
.001455604 
.001453488 
.001451379 
.001449275 


691 
692 
693 
694 
695 


477481 
478864 
480249 
481636 
483025 


329939371 
331373888 
332812557 
334255384 
335702375 


- 26.2S6« 89 
26.3058.29 
26.3248932 
26.34G8797 
26-3628527 


8-8408227 
8-8450854 
8-8493440 
8-8535985 
8-8578489 


.001447178 
.001445087 
.001443001 
.001440922 
.001438849 


696 
697 
698 
699 
700 


484416 
485809 
487204 
488601 
490000 


337153536 
338608873 
340068392 
341532099 
343000000 


26.3818119 
26.4007576 
26.4196896 
26.4386081 
26.4575131 


8-8620952 
8-8663375 
8-8705757 
8-8748099 
8.8790400 


.001436782 
.001434720 
.001432665 
.001430615 
.001428571 


701 
702 
703 
704 
705 


491401 
492804 
494209 
495616 
497025 


344472101 
345948408 
347428927 
348913664 
350402625 


26.4764046 
26.4952826 
26.5141472 
26.5329983 
26-5518361 


8-8832661 
8-8874882 
8.8917063 
8.8959204 
8-9001304 


.001426534 
.001424501 
.001422475 
.001420455 
.001418440 


706 
707 
708 
709 
710 


498436 
499849 
501264 
502681 
504100 


351895816 
353393243 
354894912 
356400829 
357911000 


26-5706605 
26-5894716 
26-6082694 
26-6270539 
26-6458252 


8.9043366 
8.9085387 
8.9127369 
8.9169311 
8-9211214 


.001416431 
.001414427 
.001412429 
.001410437 
.001408451 


711 
712 
713 
714 
715 


505521 
506944 
508369 
509796 
511225 


359425431 
360944128 
362467097 
363994344 
365525875 


26-6645833 
26-6833281 
26-7020598 
26-7207784 
26.7394839 


8.9253078 
8.9294902 
8.9336687 
8.9378433 
8-9420140 


.001406470 
.001404494 
.001402525 
.001400560 
.001398501 


716 
717 
718 
719 
730 


512656 
514089 
515524 
516961 
518400 


367061696 
368601813 
370146232 
371694959 
373248000 


26-7581763 
26.7768557 
26.7955220 
26.8141754 
26.8328157 


8-9461809 
8.9503438 
8.9545029 
8.9586581 
8.9628095 


.001396648 
.001394700 
.001392758 
.001390821 
.001388889 



757 



TABLE XXXII.-— SQUARES, CUBES, SQUARE ROOTS, 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


721 
722 
723 
724 
725 


519841 
521284 
522729 
524176 
525625 


374805361 
376367048 
377933067 
379503424 
381078125 


26.8514432 
26.8700577 
26.8886593 
26.9072481 
26.9258240 


8.9669570 
8.9711007 
8.9752406 
8.9793766 
8.9835089 


.001386963 
.001385042 
.001383126 
.001381215 
.001379310 


726 
121 
728 
729 
730 


527076 
528529 
529984 
531441 
532900 


382657176 
384240583 
385828352 
387420489 
389017000 


26.9443872 
26.9629375 
26.9814751 
27.0000000 
27.0185122 


8.9876373 
8.9917620 
8.9958829 
9.0000000 
9.0041134 


.001377410 
.001375516 
.001373626 
.001371742 
.001369863 


731 
732 
733 
734 
735 


534361 
535824 
537289 
538756 
540225 


390617891 
392223168 
393832837 
395446904 
397065375 


27.0370117 
27.0554985 
27.0739727 
27.0924344 
27.1108834 


9.0082229 
9.0123288 
9.0164309 
9.0205293 
9.0246239 


.001367989 
.001366120 
.001364256 
.001362398 
.001360544 


736 
IZl 
738 
739 
740 


541696 
543169 
544644 
546121 
547600 


398688256 
400315553 
401947272 
403583419 
405224000 


27.1293199 
27.1477439 
27.1661554 
27.1845544 
27.2029410 


9.0287149 
9.0328021 
9.0368857 
9.0409655 
9.0450419 


.001358696 
.001356852 
.001355014 
.001353180 
.001351351 


741 
742 
743 
744 
745 


549081 
550564 
552049 
553536 
555025 


406869021 
408518488 
410172407 
411830784 
413493625 


27.2213152 
27.2396769 
27.2580263 
27.2763634 
27.2946881 


9.0491142 
9.0531831 
9.0572482 
9.0613098 
9.0653677 


.001349528 
.001347709 
.001345895 
.001344086 
.001342282 


746 
747 
748 
749 
750 


556516 
558009 
559504 
561001 
562500 


415160936 
416832723 
418508992 
420189749 
421875000 


27.3130006 
27.3313007 
27.3495887 
27.3678644 
27.3861279 


9.0694220 
9.0734726 
9.0775197 
9.0815631 
9.0856030 


.001340483 
.001338688 
.001336898 
.001335113 
.001333333 


751 
752 
753 
754 
755 


564001 
565504 
567009 
568516 
570025 


423564751 
425259008 
426957777 
428661064 
430368875 


27.4043792 
27.4226184 
27.4408455 
27.45190604 
27.4772633 


9.0896392 
9.0936719 
9.0977010 
9.1017265 
9.1057485 


.001331558 
.001329787 
.001328021 
.001326260 
.001324503 


756 
757 
758 
759 
760 


571536 
573049 
574564 
576081 
577600 


432081216 
433798093 
435519512 
437245479 
438976000 


27.4954542 
27.5136330 
27.5317998 
27.5499546 
27.5680975 


9.1097669 
9.1137818 
9.1177931 
9.1218010 
9.1258053 


.001322751 
.001321004 
.001319261 
.001317523 
.001315789 


761 
762 
763 
764 
765 


579121 
580644 
582169 
583696 
585225 


440711081 
442450728 
444194947 
445943744 
447697125 


27.5862284 
27.6043475 
27.6224546 
27.6405499 
27.6586334 


9.1298061 
9.1338034 
9.1377971 
9.1417874 
9. 1457742* 


.001314060 
.001312336 
.001310616 
.001308901 
.001307190 


766 
767 
768 
769 
770 


586756 
588289 
589824 
591361 
592900 


449455096 
451217663 
452984832 
454756609 
456533000 


27.6767050 
27.6947648 
27-7128129 
27.7308492 
27-7488739 


9.1497576 
9.1537375 
9.1577139 
9.1616869 
9.1656565 


.001305483 
.001303781 
.001302083 
.001300390 
.001298701 


771 
772 
773 
774 
775 


594441 
595984 
597529 
599076 
600625 


458314011 
460099648 
461889917 
463684824 
465484375 


27.7668868 
27.7848880 
27.8028775 
27.8208555 
27. 8388218 


9.1696225 
9.1735852 
9.1775445 
9.1815003 
9.1854527 


.001297017 
.001295337 
.001293661 
.001291990 
.001290323 


776 
777 
778 
779 

780 


602176 
603729 
605284 
606841 
608400 


467288576 
469097433 
470910952 
472729139 
474552000 


27.8567766 
27.8747197 
27.8926514 
27.9105715 
27.9284801 


9.1894018 
9.1933474 
9.1972897 
9.2012286 
9.2051641 


.001288660 
.001287001 
.001285347 
.001283697 
.001282051 



758 



CUBE ROOTS, AND RElCiPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots. 


Cube Roots, 


Reciprocals. 


781 
782 
783 
784 
785 


609961 
611524 
613089 
614656 
616225 


476379541 
478211768 
480048687 
481890304 
483736625 


27.9463772 
27.9642629 
27.9821372 
28.0000000 
28.0178515 


9- 
9. 
9- 
9. 
9- 


2090962 
2130250 
2169505 
2208726 
2247914 


-001280410 
-001278772 
-001277139 
-001275510 
.001273885. 


786 
787 
788 
789 
790 


617796 
619369 
620944 
622521 
624100 


485587656 
487443403 
489303872 
491169069 
493039000 


28.0356915 
28.0535203 
28.0713377 
28.0891438 
28.1069386 


9. 

9. 

9 

9 

9 


2287068 
2326189 
2365277 
2404333 
2443355 


.001272265 
-001270648 
-001269036 
-001267427 
.001265823 


791 
792 
793 
794 
795 


625681 
627264 
628849 
630436 
632025 


494913671 
496793088 
498677257 
500566184 
502459875 


28.1247222 
28-1424946 
28.1602557 
28.1780056 
28.1957444 


9 
9 
9 
9 
9 


2482344 
2521300 
2560224 
2599114 
2637973 


.001264223 
.001262626 
.001261034 
-001259446 
-001257862 


796 
797 
798 
799 
800 


633616 
635209 
636804 
638401 
640000 


504358336 
506261573 
508169592 
510082399 
512000000 


28.2134720 
28-2311884 
28.2488938 
28.2665881 
28.2842712 


9 
9 
9 
9 
9 


2676798 
2715592 
2754352 
2793081 
2831777 


.001256281 
.001254705 
-001253133 
.001251564 
.001250000 


801 
802 
803 
804 
805 


641601 
643204 
644809 
646416 
648025 


513922401 
515849608 
517781627 
519718464 
521660125 


28.3019434 
28.3196045 
28.3372546 
28.3548938 
28-3725219 


9 
9 
9 
9 
9 


2870440 
2909072 
2947671 
2986239 
3024775 


.001248439 
o001246883 
-001245330 
-001243781 
.001242236 


806 
807 
808 
809 
810 


649636 
651249 
652864 
654481 
656100 


523606616 
525557943 
527514112 
529475129 
531441000 


28.3901391 
28-4077454 
28.4253408 
28.4429253 
28.4604989 


9 
9 
9 
9 
9 


3063278 
3101750 
3140190 
3178599 
3216975 


-001240695 
.001239157 
-001237624 
»001236094 
-001234568 


811 
812 
813 
814 
815 


657721 
659344 
660969 
662596 
664225 


533411731 
535387328 
537367797 
539353144 
541343375 


-28.4780617 
28-4956137 
28.5131549 
28.5306852 
28.5482048 


9 
9 
9 
9 
9 


3255320 
3293634 
3331916 
3370167 
3408386 


-001233046 
.001231527 
.001230012 
-001228501 
-001226994 


816 
817 
818 
819 

, 820 


665856 
667489 
669124 
670761 
672400 


543338496 
545338513 
547343432 
549353259 
551368000 


28.5657137 
28.5832119 
28.6006993 
28.6181760 
28-6356421 


9 
9 
9 
9 
9 


3446575 
3484731 
3522857 
3560952 
3599016 


-001225490 
-001223990 
-001222494 
-001221001 
-001219512 


821 
822 
823 
824 
825 


674041 
675684 
677329 
678976 
680625 


553387661 
555412248 
557441767 
559476224 
561515625 


28-6530976 
28-6705424 
28-6879766 
28-7054002 
28-7228132 


9 
9 
9 
9 
9 


3637049 
3675051 
3713022 
3750963 
3788873 


-001218027 
«001216545 
.001215067 
.001213592 
-001212121 


826 
827 
828 
829 
830 


682276 
683929 
685584 
687241 
688900 


563559976 
565609283 
567663552 
569722789 
571787000 


28.7402157 
28.7576077 
28.7749891 
28.7923601 
28-8097206 


9 
9 
9 
9 
9 


3826752 
3864600 
3902419 
3940206 
3977964 


.001210654 
-001209190 
-001207729 
-001206273 
-001204819 


831 
832 
833 
834 
835 


690561 
692224 
693889 
695556 
697225 


573856191 
575930368 
578009537 
580093704 
582182875 


28-8270706 
28-8444102 
28-8617394 
28.8790582 
28-8963666 


9 
9 
9 
9 
9 


4015691 
4053387 
4091054 
4128690 
4166297 


.001203369 
-001201923 
-001200480 
-001199041 
-001197605 


836 
837 
838 
839 
840 


698896 
700569 
702244 
703921 
705600 


584277056 
586376253 
588480472 
590589719 
592704000 


28-9136646 
28-9309523 
28-9482297 
28-9654967 
28-9827535 


9 
9 
9 
9 
9 


.4203873 
.4241420 
.4278936 
.4316423 
-4353880 


-001196172 
.001194743 
.001193317 
.001191895 
.001190476 



759 



TABLE XXXII.— SQUARES. CUBES, SQUARE ROOTS, 



No. 


Squares, 


Cubes. 


Square Roots. 


Cube Roots. 


Reciprocals. 


841 
842 
843 
844 
845 


707281 
708964 
710649 
712336 
714025 


594823321 
596947688 
599077107 
601211584 
603351125 


29.0000000 
29.0172363 
29.0344623 
29.0516781 
29.0688837 


9.4391307 
9.4428704 
9.4466072 
9.4503410 
9.4540719 


.001189061 
.001187648 
.001186240 
.001184834 
.001183432 


846 
847 
848 
849 
850 


715716 
717409 
719104 
720801 
722500 


605495736 
607645423 
609800192 
611960049 
614125000 


29.0860791 
29.1032644 
29.1204396 
29.1376046 
29.1547595 


9.4577999 
9.4615249 
9.4652470 
9.4689661 
9.4726824 


.001182033 
.001180638 
.001179245 
.001177856 
.001176471 


851 
852 
853 
854 
855 


724201 
725904 
727609 
729316 
731025 


616295051 
618470208 
620650477 
622835864 
625026375 


29.1719043 
29.1890390 
29.2061637 
29.2232784 
29.2403830 


9.4763957 
9.4801061 
9.4838136 
9.4875182 
9.4912200 


.001175088 
.001173709 
.001172333 
.001170960 
.001169591 


856 
857 
858 
859 
860 


732736 
734449 
736164 
737881 
739600 


627222016 
629422793 
631628712 
633839779 
636056000 


29.2574777 
29.2745623 
29.2916370 
29.3087018 
29.3257566 


9.4949188 
9.4986147 
, 9.5023078 
9.5059980 
9.5096854 


.001168224 
.001166861 
.001165501 
.001164144 
.001162791 


861 
862 
8&3 
864 
865 


741321 
743044 
744769 
746496 
748225 


638277381 
640503928 
642735647 
644972544 
647214625 


29.3428015 
29.3598365 
29.3768616 
29.3938769 
29.4108823 


9.5133699 
9.5170515 
9.5207303 
9.5244063 
9.5280794 


.001161440 
.001160093 
.001158749 
.001157407 
.001156069 


866 
867 
868 
869 
870 


749956 
751689 
753424 
755161 
756900 


649461896 
651714363 
653972032 
656234909 
658503000 


29.4278779 
9.4448637 
29.4618397 
29.4788059 
29.4957624 


9.5317497 
9.5354172 
9.5390818 
9.5427437 
9.5464027 


.001154734 
.001153403 
.001152074 
.001150748 
.001149425 


871 
872 
873 
874 
875 


758641 
760384 
762129 
763876 
765625 


660776311 
663054848 
665338617 
667627624 
669921875 


29.5127091 
29.5296461 
29.5465734 
29.5634910 
29.5803989 


9.5500589 
9.5537123 
9.5573630 
9.5610108 
9-5646559 


.001148106 
.001146789 
.001145475 
.001144165 
.001142857 


876 
877 
878 
879 
880 


767376 
769129 
770884 
772641 
774400 


672221376 
674526133 
676836152 
679151439 
681472000 


29.5972972 
29.6141858 
29.6310648 
29.6479342 
29.6647939 


9.5682982 
9.5719377 
9.5755745 
9.5792085 
9.5828397 


.001141553 
.001140251 
.001138952 
.001137656 
.001136364 


881 
882 
883 
884 
885 


776161 
777924 
779689 
781456 
783225 


683797841 
686128968 
688465387 
690807104 
693154125 


29.6816442 
29.6984848 
29.7153159 
29.7321375 
29.7489496 


9.5864682 
9.5900939 
9-5937169 
9.5973373 
9.6009548 


.001135074 
.001133787 
.001132503 
.001131222 
.001129944 


886 
887 
888 
889 
890 


784996 
786769 
788544 
790321 
792100 


695506456 
697864103 
700227072 
702595369 
704969000 


29.7657521 
29.7825452 
29.7993289 
29.8161030 
29-8328678 


9.6045696 
9.6081817 
9.6117911 
9-6153977 
9-6190017 


.001128668 
.001127396 
.001126126 
.001124859 
.001123596 


891 
892 
893 
894 
895 


793881 
795664 
797449 
799236 
801025 


707347971 
709732288 
712121957 
714516984 
716917375 


29.8496231 
29.8663690 
29.8831056 
29.8998328 
29-9165506 


9-6226030 
9.6262016 
9-6297975 
9-6333907 
9-6369812 


.001122334 
.001121076 
.001119821 
.001118568 
.001117318 


896 
897 
898 
899 
900 


802816 
804609 
806404 
808201 
810000 


719323136 
721734273 
724150792 
726572699 
729000000 


29-9332591 
29-9499583 
29.9666481 
29-9833287 
30.0000000 


9.6405690 
9.6441542 
9.6477367 
9.6513166 
9.6548938 


.001116071 
.001114827 
.001113586 
.001112347 
.001111111 



760 



CUBE ROOTS. AND RECIPROCALS. 



No. 


Squares. 


Cubes. 


Square Roots, 


Cube Roots. 


Reciprocals. 


901 
902 
903 
904 
905 


811801 
813604 
815409 
817216 
819025 


731432701 
733870808 
736314327 
738763264 
741217625 


30.0166620 
30.0333148 
30.0499584 
30.0665928 
30.0832179 


9.6584684 
9.6620403 
9.6656096 
9.6691762 
9.6727403 


.001109878 
.001108647 
.001107420 
.001106195 
.001104972 


906 
907 
908 
909 
910 


820836 
822649 
824464 
826281 
828100 


743677416 
746142643 
748613312 
751089429 
753571000 


30.0998339 
30.1164407 
30.1330383 
30.1496269 
30.1662063 


9.6763017 
9.6798604 
9.6834166 
9.6869701 
9.6 05211 


.001103753 
.001102536 
.001101322 
.001100110 
.001098901 


911 
912 
913 
914 
915 


829921 
831744 
833569 
835396 
837225 


756058031 
758550528 
761048497 
763551944 
766060875 


30.1827765 
30.1993377 
30.2158899 
30.2324329 
30.2489669 


9.6940694 
9.6976151 
9.7011583 
9.7046989 
9.7082369 


.001097695 
.001096491 
.001095290 
.001094092 
.001092896 


916 
917 
918 
919 
920 


839056 
840889 
842724 
844561 
846400 


768575296 
771095213 
773620632 
776151559 
778688000 


30.2654919 
30.2820079 
30.2985148 
30.3150128 
30.3315018 


9.7117723 
9.7153051 
9.7188354 
9-7223631 
9.7258883 


.001091703 
.001090513 
.001089325 
.001088139 
.001086957 


921 
922 
923 
924 
925 


848241 
850084 
851929 
853776 
855625 


781229961 
783777448 
786330467 
788889024 
791453125 


30.3479818 
30.3644529 
30.3809151 
30.3973683 
30.4138127 


9.7294109 
9.7329309 
9.7364484 
9.7399634 
9.7434758 


.001085776 
.001084599 
.001083423 
.001082251 
.001081081 


926 
927 
928 
929 
930 


857476 
859329 
861184 
863041 
864900 


794022776 
796597983 
799178752 
801765089 
804357000 


30.4302481 
30.4466747 
30.4630924 
30.4795013 
30.4959014 


9-7469857 
9-7504930 
9-7539979 
9.7575002 
9.7610001 


.001079914 
.001078749 
.001077586 
.001076426 
.001075269 


931 
932 
933 
934 
935 


866761 
868624 
870489 
872356 
874225 


806954491 
809557568 
812166237 
814780504 
817400375 


30-5122926 
30.5286750 
30.5450487 
30.5614136 
30.5777697 


9.7644974 
9.7679922 
9.7714845 
9.7749743 
9.7784616 


.001074114 
.001072961 
.001071811 
.001070664 
.001069519 


936 
937 
938 
939 
940 


876096 
877969 
879844 
881721 
883600 


820025856 
822656953 
825293672 
827936019 
830584000 


30.5941171 
30.6104557 
30.6267857 
30.6431069 
30.6594194 


9.7819466 
9.7854288 
9.7889087 
9.7923861 
9.7958611 


.001068376 
.001067236 
.001066098 
.001064963 
.001063830 


941 
942 
943 
944 
945 


885481 
887364 
889249 
891136 
893025 


833237621 
835896888 
838561807 
841232384 
843908625 


30.6757233 
30.6920185 
30.7083051 
30.7245830 
30.7408523 


9.7993336 
9.8028036 
9.8062711 
9.8097362 
9.8131989 


.001062699 
.001061571 
.001060445 
.001059322 
.001058201 


946 
947 
948 
949 
950 


894916 
896809 
898704 
900601 
902500 


846590536 
849278123 
851971392 
854670349 
857375000 


30.7571130 
30.7733651 
30.7896086 
30.8058436 
30.8220700 


9.8166591 
9.8201169 
9.8235723 
9.8270252 
9.8304757 


.001057082 
.001055966 
.001054852 
.001053741 
.001052632 


951 
952 
953 
954 
955 


904401 
906304 
908209 
910116 
912025 


860085351 
862801408 
865523177 
868250664 
870983875 


30.8382879 
30.8544972 
30.8706981 
30.8868904 
30.9030743 


9.8339238 
9.8373695 
9.8408127 
9.8442536 
9.8476920 


.001051525 
.001050420 
.001049318 
.001048218 
.001047120 


956 
957 
958 
959 
960 


913936 
915849 
917764 
919681 
921600 


873722816 
876467493 
879217912 
881974079 
884736000 


30.9192497 
30.9354166 
30.9515751 
30.9677251 
80.9838668 


9.8511280 
9.8545617 
9.8579929 
9.8614218 
9.8648483 


.001046025 
.001044932 
.001043841 
.001042753 
.001041667 



761 





TABLE XXXII.- 


-SQUARES, CUBES, ETC. 


'»' 


No. 


Squares, 


Cubes. 


Square Roots. 


Cube Roots. 


Recipsocals. \ 


961 


923521 


887503681 


31.0000000 


9.8682724 


.001040583 


962 


925444 


890277128 


31.0161248 


9.8716941 


.001089501 


963 


927369 


893056347 


31.0322413 


9.8751135 


.001038422 


964 


929296 


895841344 


31.0483494 


9.8785305 


.001037344 i 


965 


931225 


898632125 


31.0644491 


9.8819451 


.001036269 i 


966 


933156 


901428696 


31.0805405 


9.8853574 


.001035197 


967 


935089 


904231063 


31.0966236 


9.8887673 


.001034126 


968 


937024 


907039232 


31.1126984 


9.8921749 


.001033058 


969 


938961 


909853209 


31.1287648 


9.8955801 


.001031992 


970 


940900 


912673000 


31.1448230 


9.8989830 


.001030928 


971 


942841 


915498611 


31.1608729 


9.9023835 


.001029866 


972 


944784 


918330048 


31.1769145 


9.9057817 


.001028807 


973 


946729 


921167317 


31.1929479 


9.9091776 


.001027749 


974 


948676 


924010424 


31.2089731 


9.9125712 


.001026694 


975 


950625 


926859375 


31.2249900 


9.9159624 


.001025641 


976 


952576 


929714176 


31.2409987 


9.9193513 


.001024590 


977 


954529 


932574833 


31.2569992 


9.9227379 


.001023541 


978 


956484 


935441352 


31.2729915 


9.9261222 


.001022495 


979 


958441 


938313739 


31.2889757 


9.9295042 


.001021450 


980 


960400 


941192000 


31.3049517 


9.9328839 


.001020408 


981 


962361 


944076141 


31.3209195 


9.9362613 


.001019368 


982 


964324 


946966168 


31.3368792 


9.9396363 


.001018330 


983 


966289 


949862087 


31.3528308 


9.9430092 


.001017294 


984 


968256 


952763904 


31.3687743 


9.9463797 


.001016260 


985 


970225 


955671625 


31.3847097 


9.9497479 


.001015228 


986 


972196 


958585256 


31.4006369 


9.9531138 


.001014199 


987 


974169 


961504803 


31.4165561 


9.9564775 


.001013171 


988 


976144 


964430272 


31.4324673 


9.9598389 


.001012146 


989 


978121 


967361669 


31.4483704 


9.9631981 


.001011122 


990 


980100 


970299000 


31.4642654 


9.9665549 


.001010101 


991 


982081 


973242271 


31.4801525 


9.9699095 


.001009082 


992 


984064 


976191488 


31.4960315 


9.9732619 


.001008065 


993 


986049 


979146657 


31.5119025 


9.9766120 


.001007049 


994 


988036 


982107784 


31.5277655 


9.9799599 


.001006036 


995 


990025 


985074875 


31.5436206 


9.9833055 


.001005025 


996 


992016 


988047936 


31.5594677 


9-9866488 


.001004016 


997 


994009 


991026973 


31.5753068 


9.9899900 


.001003009 


998 


996004 


994011992 


31.5911380 


9.9933289 


.001002004 


999 


998001 


997002999 


31.6069613 


9.9966656 


.001001001 


1000 


1000000 


1000000000 


31.6227766 


10.0000000 


.001000000 


1001 


1002001 


1003003001 


31.6385840 


10.0033322 


.0009990010 


1002 


1004004 


1006012008 


31.6543836 


10.0066622 


.0009980040 


1003 


1006009 


1009027027 


31.6701752 


10.0099899 


.0009970090 


1004 


1008016 


1012048064 


31.6859590 


10.0133155 


.0009960159 


1005 


1010025 


1015075125 


31.7017349 


10.0166389 


.0009950249 


1006 


1012036 


1018108216 


31.7175030 


10.0199601 


.0009940358 


1007 


1014049 


1021147343 


31.7332633 


10.0232791 


.0009930487 


1008 


1016064 


1024192512 


31.7490157 


10.0265958 


.0009920635 


1009 


1018081 


1027243729 


31.7647603 


10.0299104 


.0009910803 


1010 


1020100 


1030301000 


31.7804972 


10.0332228 


.0009900990 


1011 


1022121 


1033364331 


31.7962262 


10.0365330 


.0009891197 


1012 


1024144 


1036433728 


31.8119474 


10.0398410 


.0009881423 


1013 


1026169 


1039509197 


31.8276609 


10.0431469 


.0009871668 


1014 


1028196 


1042590744 


31.8433666 


10.0464506 


.0009861933 


1015 


1030225 


1045678375 


31.8590646 


10.0497521 


.0009852217 


1016 


1032256 


1048772096 


31.8747549 


10.0530514 


.0009842520 


1017 


1034289 


1051871913 


31.8904374 


10.0563485 


.0009832842 


1018 


1036324 


1054977832 


31.9061123 


10.0596435 


.0009823183 


1019 


1038361 


1058089859 


31.9217794 


10.0629364 


.0009813543 


1020 


1040400 


1061208000 


31.9374388 


10.0662271 


.0009803922 



762 



TABLE XXXIII.— CUBIC YARDS PER 100 FEET OF LEVEL 
SECTIONS. SLOPE 1:1. 



Depth, 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


d 


12 feet. 


14 feet. 


16 feet. 


18 feet. 


20 feet. 


28 feet. 


30 feet. 


32 feet. 


1 


48 


56 


63 


70 


78 


107 


115 


122 


2 


104 


119 


133 


148 


163 


222 


237 


252 


S 


167 


189 


211 


233 


256 


344 


367 


389 


4 


237 


267 


296 


326 


356 


474 


504 


533 


5 


315 


352 


389 


426 


463 


611 


648 


685 


6 


400 


444 


489 


533 


578 


756 


800 


844 


7 


493 


544 


596 


648 


700 


907 


959 


1011 


8 


593 


652 


711 


770 


830 


1067 


1126 


1185 


9 


700 


767 


833 


900 


967 


1233 


1300 


1367 


10 


815 


889 


963 


1037 


1111 


1407 


1481 


1556 


11 


937 


1019 


1100 


1181 


1263 


1589 


1670 


1752 


12 


1067 


1156 


1244 


1333 


1422 


1778 


1867 


1959 


13 


1204 


1300 


1396 


1493 


1589 


1974 


2070 


216? 


14 


1348 


1452 


1556 


1659 


1763 


2178 


2281 


2385 


15 


1500 


1611 


1722 


1833 


1944 


2389 


2500 


2611 


16 


1659 


1778 


1896 


2015 


2133 


2607 


2726 


2844 


17 


1826 


1952 


2078 


2204 


2330 


2833 


2959 


3085 


18 


2000 


2133 


2267 


2400 


2533 


3067 


3200 


3333 


19 


2181 


2322 


2463 


2604 


2744 


3307 


3448 


3589 


20 


2370 


2519 


2667 


2815 


2963 


3556 


3704 


3852 


21 


2567 


2722 


2878 


3033 


3189 


3811 


3967 


4122 


22 


2770 


2933 


3096 


3259 


3422 


4074 


4237 


4400 


23 


2981 


3152 


3322 


3493 


3663 


4344 


4515 


4685 


24 


3200 


3378 


3556 


3733 


3911 


4622 


4800 


4978 


25 


3426 


3611 


3796 


3981 


4167 


4907 


5093 


5278 


26 


3659 


3852 


4044 


4237 


4430 


5200 


5393 


5585 


27 


3900 


4100 


4300 


4500 


4700 


5500 


5700 


5900 


28 


4148 


4356 


4563 


4770 


4978 


5807 


6015 


6222 


29 


4404 


4619 


4833 


5048 


5263 


6122 


6337 


6552 


30 


4667 


4889 


5111 


5333 


5556 


6444 


6667 


6889 


31 


4937 


5167 


5396 


5626 


5856 


6774 


7004 


7233 


32 


5215 


5452 


5689 


-5926 


6163 


7111 


7348 


7585 


33 


5500 


5744 


5989 


6233 


6478 


7456 


7700 


7944 


34 


5793 


6044 


6296 


6548 


6800 


7807 


8059 


8311 


35 


6093 


6352 


6611 


6870 


7130 


8167 


8426 


8685 


36 


6400 


6667 


6933 


7200 


7467 


8533 


8800 


9067 


g7 


6715 


6989 


7263 


7537 


7811 


8907 


9181 


9456 


38 


7037 


7319 


7600 


7881 


8163 


9289 


9570 


9852 


39 


7367 


7656 


7944 


8233 


8522 


9678 


9967 


10256 


40 


7704 


8000 


8296 


8593 


8889 


10074 


10370 


10667 


41 


8048 


8352 


8656 


8959 


9263 


10478 


10781 


11085 


42 


8400 


8711 


9022 


9333 


9644 


10889 


11200 


11511 


43 


8759 


9078 


9396 


9715 


10033 


11307 


11626 


11944 


44 


9126 


9452 


9778 


10104 


10430 


11733 


12059 


12385 


45 


9500 


9833 


10167 


10500 


10833 


12167 


12500 


12833 


46 


9881 


10222 


10563 


10904 


11244 


12607 


12948 


13289 


47 


10270 


10619 


10967 


11315 


11663 


13056 


13404 


13752 


48 


10667 


11022 


11378 


11733 


12089 


13511 


13867 


14222 


49 


11070 


11433 


11796 


12159 


12522 


13974 


14337 


14700 


50 


11481 


11852 


12222 


12593 


12963 


14444 


14815 


15185 


51 


11900 


12278 


12656 


13033 


13411 


14922 


15300 


15678 


52 


12326 


12711 


13096 


13481 


13867 


15407 


15793 


16178 


53 


12759 


13152 


13544 


13937 


14330 


15900 


16293 


16685 


54 


13200 


13600 


14000 


14400 


14800 


16400 


16800 


17200 


55 


13648 


14056 


14463 


14870 


15278 


16907 


17315 


17722 


56 


14104 


14519 


14933 


15348 


15763 


17422 
17944 


17837 


18252 


57 


14567 


14989 


15411 


15833 


16256 


18367 


18789 


58 


15037 


15467 


15896 


16326 


16756 


18474 


18904 


19333 


59 


15515 


15952 


16389 


16826 


17263 


19011 


19448 


19885 


60 


16000 


16444 


16889 


17333 


17778 


19556 


20000 


20444 



763 



TABLE XXXIII.— CUBIC YARDS PER 100 FEET OF LEVEL 






SECTIONS. SLOPE 1.5; 


1. 






Depth 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


Base 


12 feet. 


14 feet. 


16 feet. 


18 feet. 


20 feet. 


28 feet. 


30 feet. 


32 feet. 


1 


50 


57 


65 


72 


80 


109 


117 


124 




111 


126 


141 


156 


170 


230 


244 


259 


3 


183 


206 


228 


250 


272 


361 


383 


406 


4 


267 


296 


326 


356 


385 


504 


533 


563 


6 


361 


398 


435 


472 


509 


657 


694 


731 


6 


467 


511 


556 


600 


644 


822 


867 


911 


? 


583 


635 


687 


739 


791 


998 


1050 


1102 


8 


711 


770 


830 


889 


948 


1185 


1244 


1304 


9 


850 


917 


983 


1050 


1117 


1383 


1450 


1517 


10 


1000 


1074 


1148 


1222 


1296 


1593 


1667 


1741 


11 


1161 


1243 


1324 


1406 


1487 


1813 


1894 


1976 


12 


1333 


1422 


1511 


1600 


1689 


2044 


2133 


2222 


13 


1517 


1613 


1709 


1806 


1902 


2287 


2383 


2480 


14 


1711 


1815 


1919 


2022 


2126 


2541 


2644 


2748 


15 


1917 


2028 


2139 


2250 


2361 


2806 


2917 


3028 


16 


2133 


2252 


2370 


2489 


2607 


3081 


3200 


3319 


17 


2361 


2487 


2613 


2739 


2365 


3369 


3494 


3620 


18 


2600 


2733 


2867 


3000 


3133 


3667 


3800 


3933 


19 


2850 


2991 


3131 


3272 


3413 


3976 


4117 


4257 


20 


3111 


3259 


3407 


3556 


3704 


4296 


4444 


4593 


21 


3383 


3539 


3694 


3850 


4006 


4628 


4783 


4939 


22 


3667 


3830 


3993 


4156 


4319 


4970 


5133 


5296 


23 


3961 


4131 


4302 


4472 


4642 


5324 


5494 


5665 


24 


4267 


4444 


4622 


4800 


4978 


5689 


5867 


6044 


25 


4583 


4769 


4954 


5139 


5324 


6065 


6250 


6435 


26 


4911 


5104 


5296 


5489 


5681 


6452 


6644 


6837 


27 


5250 


5450 


5650 


5850 


6050 


6850 


7050 


7250 


28 


5600 


5807 


6015 


6222 


6430 


7259 


7467 


7674 


29 


5961 


6176 


6391 


6606 


6820 


7680 


7894 


8109 


80 


6333 


6556 


6778 


7000 


7222 


8111 


8333 


8556 


81 


6717 


6946 


7176 


7406 


7635 


8554 


8783 


9013 


32 


7111 


7348 


7585 


7822 


8059 


9007 


9244 


9481 


83 


7517 


7761 


8006 


8250 


8494 


9472 


9717 


9961 


84 


7933 


8185 


8437 


8689 


8941 


9948 


10200 


10452 


85 


8361 


8620 


8880 


9139 


9398 


10435 


10694 


10954 


86 


8800 


9067 


9333 


9600 


9867 


10933 


11200 


11467 


87 


9250 


9524 


9798 


10072 


10346 


11443 


11717 


11991 


88 


9711 


9993 


10274 


10556 


10837 


11963 


12244 


12526 


89 


10183 


10472 


10761 


11050 


11339 


12494 


12783 


13072 


40 


10667 


10963 


11259 


11556 


11852 


13037 


13333 


13630 


41 


11161 


11465 


11769 


12072 


12376 


13591 


13894 


14198 


42 


11667 


11978 


12289 


12600 


12911 


14156 


14467 


14778 


43 


12183 


12502 


12820 


13139 


13457 


14731 


15050 


15369 


44 


12711 


13037 


13363 


13689 


14015 


15319 


15644 


15970 


45 


13250 


13583 


13917 


14250 


14583 


15917 


16250 


16583 


46 


13800 


14141 


14481 


14822 


15163 


16526 


16867 


17207 


47 


14361 


14709 


15057 


15406 


15754 


17146 


17494 


17843 


48 


14933 


15289 


15644 


16000 


16356 


17778 


18133 


18489 


49 


15517 


15880 


16243 


16606 


16969 


18420 


18783 


19146 


50 


16111 


16481 


16852 


17222 


17593 


19074 


19444 


19815 


61 


16717 


17094 


17472 


17850 


18228 


19739 


20117 


20494 


52 


17333 


17719 


18104 


18489 


18874 


20415 


20800 


21185 


53 


17961 


18354 


18746 


19139 


19531 


21102 


21494 


21887 


54 


18600 


19000 


19400 


19800 


20200 


21800 


22200 


22600 


55 


19250 


19657 


20065 


20472 


20880 


22509 


22917 


23324 


56 


19911 


20326 


20741 


21156 


21570 


23230 


23644 


24059 


57 


20583 


21006 


21428 


21850 


22272 


23961 


24383 


24805 


58 


21267 


21696 


22126 


22556 


22985 


24704 


25133 


25563 


59 


21961 


22398 


22835 


23272 


23709 


25457 


25894 


26331 


60 


22667 


23111 


23556 


24000 


24444 


26222 


26667 


27111 



764 



TABLE 



XXXIII.— CORRECTIVE PERCENTAGE 
TABLES OF LEVEL SECTIONS. 



To be applied when cross-sections are not level. 
Side slope = 1.5:1 or /? = 33^41'. 



FACTORS FOR 
See § 95. 



Trans- 
verse 

surface 
slope. 


6=12 feet 
and d= 


6=20 feet 
and d= 


6=30 feet 
and d= 


a° 


Per- 
cent 


10 

feet. 


20 

feet. 


50 
feet. 


10 
feet. 


20 
feet. 


50 

feet. 


10 

feet. 


20 
feet. 


50 

feet. 


5 

10 
15 
20 
30 


9 
18 
27 
36 
57 


8.2 
21 

4e 

327 


% 

1.8 

7.7 

20 

44 
324 


r»8 

7.5 

19 

43 

317 


9.0 

23 

51 

358 


7o 

1.8 

8.0 

21 

45 
336 


7.6 
20 
44 
321 


% 
2.3 
10.0 
26 
57 
400 


8.4 

22 

48 

354 


% 

1.8 

7.7 

20 

44 
326 



Sideslope=l:l or/? =45° 



Trans- 
verse 

surface 
slope. 


6=12 feet 
andd= 


6=20 feet 
and d= 


6=30 feet 
and d= 


aP 


Per- 
cent 


10 

feet. 


20 
feet. 


50 

feet. 


10 

feet. 


20 
feet. 


50 
feet. 


10 
feet. 


20 
feet. 


50 
feet. 


5 
10 
15 
20 
30 


9 
18 

27 
36 
57 


% 
0.9 
3.7 
9.0 

18 

58 


0^!8 
3.4 
8.2 

16 

53 


% 
0.8 
3.2 
7.8 

15 

50 


- 4.3 
10.3 
20 
67 


% 
0.9 
3.6 
8.7 

17 

56 


% 
0.8 
3.3 
8.0 

16 

51 


5.0 
12.1 
24 
78 


% 
0.9 
4.0 
9.5 

19 

61 


% 
0.8 
3.4 
8.2 

16 

53 



765 






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766 



INDEX. 

Numbers refer to sections except where specifically marked pages (p.). 

Abandonment of existing track , 457, c. 

Abutments for trestles 142 

Accelerated motion, application of laws to movement of trains 431 

Accidents, da-nger of, due to curvature 418 

Accuracy of earthwork computations , 94 

nmnerical example 86 

tunnel surveying 163 

Additional business ; methods of securing (or losing) it 455 

train to handle a given traffic, cost of — Table XXVII p. 532 

Adhesion of wheels and rails 333, 334 

Adjustments of dumpy level — Appendix pp. 561, 562 

instruments, general principles — Appendix p. 554 

transit — Appendix pp. 555-559 

wye level — Appendix • • . • .pp. 559-"S6x 

Advance signals, in block signaling , 307 

Advantages of re-location of old lines , 456 

tie-plates 244 

Air-brakes 336, 337 

Air resistance — see Atmospheric resistance. 

Allowance for shrinkage of earthwork 97 

American locomotives, frame 316 

equalizing levers 324 

running gear 323 

system of tunnel excavation 171 

Aneroid barometer, use in reconnoissance leveling 7 

Angle-bars, cost 358, d 

efficiency of 238 

number per mile of track — Table XVII. ...p. 432 

standard 242 

Angle of slope in earthwork 60 

Annual charge against a tie, at 5% interest — ^Table XXXIV p. 766 

Apprehension of danger, effect on travel 419, e 

ARCH CULVERTS 191-192 

design 191 

example 192 

Area of culverts, computation 178-183 

A. S. C. E. standard rail sections 226 

Assistant engines — see Pusher engines and Pusher grades. 

Atmospheric resistance, train 342 

767 



768 INDEX. r- 

Atlantic locomotives, running gear 323 

Austrian system of tunnel excavation 171 

Automatic air-brakes 337 

signaling, track circuit 310 

Averaging end areas, volume of prismoid computed by 72 

Axle, effect of parallelism 312 

effect of rigid wheels on 311 

radial, possibilities of » 313 

size of standard M.C.B 332 

Balance of grades for unequal traflSc 452-454 

determination of relative traffic . . . 454 

general principle 452 

theoretical balance 453 

Balanced grades for one, two, and three engines — Table XXVIII. ... p. 536 

Baldwin Locomotive Works formula for train resistance 350 

BALLAST.— Chap. VII. 

cost 200, 358, a 

cross-sections 198 

laying. = 199 

materials , 197 

Banjo signals, in block signalling 308 

Barometer, reduction of readings to 32*^ F. — Table XI p. 741 

use of aneroid in reconnoissance leveling.. 7 

Barometric elevations — Table XII. p. 742 

coefficients for corrections for temperatures and 

humidity — Table XIII. p. 742 

Beams, strength of stringers considered as , 156 

Bearings, compass, use as check on deflections, 16, 17 

in preliminary surveys 11 

Belgian system of tunnel excavation . . 171 

Belpaire fire-box. 318 

Blasting 117-123 

use in loosening earth 107, c 

BLOCK SIGNALING.— Chap. XIV. 

Boiler for locomotive 317, 318 

Boiler-power of locomotives, relation to tractive and cylinder power. . 326 
Bolts — see Track bolts. 

Bonds of railroads, security and profits 369 

Borrow-pits, earthwork 89 

Bowls (or pots) as rail supports 201, 223 

Box-cars, size and capacity 328 

Box culverts 188-190 

old-rail 190 

stone 189 

wooden 188 

Bracing for trestles 140, 141 

design • 159 

Brakes — see Train-brakes. 

Brake resistances 346 

Bridge joints (rail) 240 

'6tid£0 spirals* •••••••••••• • • 5 



INDEX. 769 

Bridges and culverts, as affected by changes in alignment 405, 422 

437, 444. 450 

cost of repairs and renewals 386 

Bridges of standard dimensions for small spatks 195 

in block signaling 308 

Bridges, trestles, and culverts on railroads, cost 357 

Broken-stone ballast 197 

Bumettizfng (chloride-of-zinc process) for preserving timber 213 

Burnt clay ballast 197 

Capital, railroad, classification of 369 

returns on 369, 370 

Caps (trestle), design 158 

Car mileage, nature and cost — Table XX, p. 470-473 » and 402 

Cars 328-332 

brake-beams , 330 

capacity and size ; 328 

causes of deterioration ^ 406 

cost of renewals and repairs 391 

as affected by changes in alignment . . 406, 
423, 437, 444, 450 

draft-gear 331 

gauge of wheel and form of wheel-tread 332 

stresses in car frames 329 

truck frames 330 

use of metal 330 

wheels, kinetic energy of 347 

Cars and horses, use in earthwork 109, e 

and locomotives, use in earthwork 109, / 

Carts and horses, use in earthwork^^. 109, a 

Cattle guards 193 

passes 194 

Centre of gravity of side-hill sections, earthwork 92 

Central angle of a curve 21 

Centrifugal force, counteracted by superelevation of outer rail 41, 42 

of connecting-rod, etc., of locomotive 325 

Chairs as supports for double-headed rails 226 

Chats for ballast 197 

Chemical composition of rails 232, 233 

purification of water 281 

Chert for ballast 197 

Cinders for ballast 197 

Circular lead rails for switches 262 

Clark's formula for train resistance 350 

Classification of excavated material 124 

Clearance card in permissive. block signaling 304 

spaces in locomotives 321 

Clearing and grubbing for railroads, cost 355 

Coal consumption in locomotives 319 

per car-mile 319 

Columbia locomotives, running gear '. 323 

Compass, use of, in preliminary surveys 11 

Competitive traffic 409 et seq. 



770 INDEX. -^ ^ ^ 

Competitive rates, equality, regardless of distance 410 

Compensation for curvature „ , . 427, 428 

Compensation for curvature rate , 428 

reasons , , . 427 

rules for 428 

Compensators in block signaling 309 

Compound curves 37-40 

modifications of location ,, 39 

nature and use , 37 

multiform, used as transition curves 45 

mutual relations of the parts 38 

Compound sections, earthwork , 61 

Computation of earthwork 71-95 

approximate, from profiles 95 

using a slide rule 80 

Conducting transportation, cost of . . . , 394-402 

as affected by changes in curvature 424 

distance 407 

minor grades .... 437 

ruling grades 444 

pusher engines 450 

Coning wheels, effect 313 

Connecting curve from a curved track to the inside 273 

from a curved track to the outside 272 

from a straight track 271 

Consolidation locomotives, equalizing levers 324 

frame 316 

rimning gear 323 

Constants, numerical, in common use — Table XXXI p. 745 

Construction of tunnels • 169-174 

Contours, obtained by cross-sectioning 12 

Contractors profit, earthwork. 115 

Corbels for trestles » . . . . 144 

Cost of an additional train to handle a given traffic — Table XXVII . . . 445 

of ballast 200 

of blasting 123 

of chemical treatment of timber ,. 216 

of earthwork 106 et seq. 

of framed-timber trestles 150 

of metal ties • 222 

of pile trestles • 134 

OF RAILROADS.— Chap. XVII. 

detailed estimate ,«., 862 

of rails 236 

of ties 209 

of transportation 364 

of treating wooden ties 216 

of tunneling 1-5 

Counterbalancing for locomotives. 325 

Crawford's formula for train resistance 350 

Creosoting for preserving timber 212 

Cross-country routes — reconnoissance 4 



INDEX. 771 

Crossings, one straight, one curved track 278 

two curved tracks 279 

Crossings, two curved tracks, numerical example 279 

two straight tracks 277 

Cross-over between two parallel curved tracks, reversed curve 275, b 

curved tracks, straight connecting 

curve 275, a 

straight tracks 274 

Cross-sectioning, for earthwork computations 68 

for preliminary surveys 12 

irregular sections for earthwork computations 87 

Cross^sections of ballast 198 

of tunnels 164 

Cross-ties — see Ties. 

Crown-bars in locomotive fire-box 318 

Cubic yards per 100 feet of level sections — ^Table XXXIII .... pp. 763-765 
CULVERTS AND MINOR BRIDGES.— Chap. VI. 

Culverts, arch 191, 192 

area of waterway 178-183 

iron-pipe 186 

old-rail 190 

reinforced-concrete * 190, a 

stone box , 189 

tile-pipe 187 

wooden box 188 

CURVATURE.— Chap. XXII. 

compensation for 427, 428 

correction for, in earthwork computations 90-93 

danger of accident due to 418 

effect on cost of conducting transportation 424 

of maintenance of equipment o . . 423 

of maintenance of way 422 

operating expenses of a change of 1 ° — Table XXII , 425 

^' travel 419 

extremes of sharp 429 

general objections 417 

of existing track, determination 35 

proper rate of compensation 428 

Curve, elements of a !• 23 

location by deflections 25 

by middle ordinates 29 

by offsets from long chord 30 

by tangential offsets 28 

by two transits , 27 

resistance of trains 311, 312, 345 

effect on cost of conducting transportation . . 424 
maintenance of equipment. . 423 

maintenance of way 422 

Curves, elements of » 21 

instrumental work in location 26 

limitations in location , 34 

method of computing length o . . . . 20 

modifications of location •••••^ 33 

mutual relations of elements ••••»»•«••••••• 22 



772 INDEX. 

Curves, obstacles to location .••.•. 32 

simple, method of designation 18 

use and value of other methods of location (not using a transit). . 31 

Cylinder power of locomotives, relation to boiler and tractive power. . . 326 

Deflecting rods for operating block signals 309 

Deflections for a spiral 48 

Degree of a curve 18 

Design of culverts 177 et seq, 

framed trestles 151-159 

bracing 159 

caps and sills . , , 158 

floor stringers 156 

posts 157 

nutlocks 253 

pile trestles 133 

tie-plates 245 

track bolts 252 

tunnels 168 

distinctive systems 171 

Development, definition , . , 6 

example, with map 6 

methods of reducing grade 5 

Disadvantages of re-locations of old lines 457 

DISTANCE.— Chap. XXI. 

effect of change on business done 416 

on division of through rates 411 

effect on operating expenses 405- 408 

Justification of decrease to save time 415 

relation to rates and expenses 403 

Distant signals in block signaling 306 

Ditches to drain road-bed 64 

Dividends actually paid on railroad stock 369 

Double-ender locomotives, running gear .••••... 323 

Double-track, distance between centers 62 

Draft gear = . 331 

"continuous" 331 

Drainage of road-bed, value of 64, 65 

Drains in tunnels • 168 

Draw -bars 331 

Drilling holes for blasting. 118, 119 

Driving-wheels of locomotives 323 

section of 325 

Drop tests for train resistance 349 

Durability of metal ties 219 

rails 234, 235 

wooden ties 204 

Dynamometer tests of train resistance 348 

Earnings of railroads, estimation of 373 

per mile of road 373 

EARTHWORK.— Chap. III. 

Earthwork computations, accuracy 94 

approximate computations from profiles.., 96 

probable error ••..•••••• • • • • 85 



INDEX. 773 

Earthwork computations, relation of actual volume to numerical result 66 

Earthwork, cost 106 et seq., 356 

limit of free haul 105 

method of computing haul 99 e^ «e^. 

shrinkage ^ . , 96 

surveys 66-70 

Eccentricity of center of gravity of earthwork cross-section 91 

Economics, railroad, justification of methods of computation 367,426 

nature and limitations 365 

of ties 202 

of treated ties 217 

Elements of a 1® curve 23 

simple curve 21 

transition cin-ves — Table IV pp. 573-58 1 

Embankments, method of formation 98 

usual form of cross-section , 58 

Empirical formulae for culvert area 180 

accuracy required 183 

value 181 

Engineering, proportionate and actual cost, in railroad construction. . 353 

Engineering News formula for pile-driving 131 

for train resistance 350 

Engineer's duties in locating a railroad 366 

Engine-houses for locomotives 289 

Enginemen, basis of wages 396, 407 

English system of tunnel excavation 171 

Enlargement of tunnel headings 170 

Entrained water in steam 321 

Equalizing-levers on locomotives. ^, 324 

Equipment, effect of curvature on maintenance of 423 

Equivalent level sections in earthwork, determination of area 78 

sections in earthwork, determination of area 77 

Estimation of probable volume of traffic and of probable growth 373 

Excavation, usual form of cross-section 58 

Exhaust-steam, effect of back-pressm-e 321 

Expansion of rails 230 

Explosives, amount used 120 

firing 122 

tamping. » 121 

use in blasting 117 

Expenditure of money for railroad purposes, general principles 377 

External distance, simple curve 2L 

table of, for a 1° curve — Table II p. 568 

Factors of safety, design of timber trestles 155 

Failures of rail joints 241 

Fastenings for metal cross-ties 221 

Field work for locating a simple curve 26 

a spiral 50 

Fire-box of locomotive 318 

required area. •••.. 318 

Fire-brick arches in locomotive fire-box 318 



774 INDEX. 

Fire protection on trestles 148 

Five-level sections in earthwork, computation of area 81 

Fixed charges, nature and ratio to total disbursements 378 

Flanges of wheels, form 332 

Flanging locomotive driving-wheels, effect 314 

Floor systems for trestling 143-150 

Formation of embankments, earthwork 96-98 

railroad corporations, method 368 

Fonnulae for pile-driving 131 

required area of culverts 180 

train resistance 350 

trigonometrical — Table XXX pp. 743-744 

useful, and constants — Table XXXI p. 74s 

Forney's formula for train resistance 350 

Fouling point of a siding 310 

Foundations for framed trestles 139 

FRAMED TRESTLES 135-159 

abutments o 142 

bracing 140, 141 

cost 150 

design 135, 151-159 

foundations 139 

joints 136 

multiple story construction 137 

span 138 

Frame of locomotive, construction. 316 

Free haul of earthwork, limit of 105 

Freight yards 295-299 

general principles 295 

minor yards 297 

relation of yard to main track 296 

track scales 299 

transfer cranes 288 

French system of tunnel excavation 171 

Friction, laws of, as applied to braking trains 334 

Frogs, diagrammatic design 255 

EUiot, illustrated in Plate VIII, p. 291. 

for switches 255, 256 

to find frog number c 256 

trigonometrical functions — Table III, p. 571. 
Weir, illustrated in Plate VIII, p. 291. 

Fuel for locomotives, cost of 397, 407, item 82. and Table XX 

as affected by changes in alignment, 407, 424, 

437. 444, 450 

German system of tunnel excavation 171 

GRADE.— Chap. XXIII. 

(see Minor grades, Pusher grades. Ruling grades.) 

accelerated motion ot trains on <, 431 

distinction between ruling and minor grades 430 

in tunnels 165 

line, change in, based on mass diagram 104 

resistance of • 344 



INDEX. 775 

Grade, undulatory, advantages, disadvantages, and safe limits 434 

virtual 432 

use, value, and misuse 433 

Grade resistance of trains 344 

Gravel ballast. I97 

Gravity tests of train resistance 349 

Grate area of locomotives 318, 320 

ratio to total heating surface 320 

Gravity, effect on trains on grades 344 

tests of train resistance 349 

Ground levers for switches 259 

Growth of railroad traffic 373 

affected by increase of facilities 375 

Guard rails for switches 261 

for trestles 145 

Guides around curves and angles (signaling mechanism) 309 

Gun-powder pile-drivers 130 

Hand-brakes 335 

Haul of earthwork, computation of length 99 et seq. 

cost 109, 116, 124, 125 

limit of profitable. 116 

method depending on distance hauled 110 

Headings in tunnels 169 

Heating surface in locomotives 320 

Henderson's formula for train resistance 350 

Hoosac Tunnel, surveys for 160, 163 

I-beam bridges, standard 195 

IMPROVEMENT OF OLD LINES.— Chap. XXIV. 

classification 455 

Inertia resistances , T. 347 

Instrumental *7ork in locating simple curves 26 

spirals 50 

Interest on cost of railroads during construction 360 

Iron pipe culverts 186 

Irregular prismoids, volume 83 

numerical example 84 

sections in earthwork, computation of area 83 

Joints, framed trestles 136 

rail 237-243 

Journal friction of axles 343, b 

Kinetic energy of trains 431 

Kyanizing (bichlorlde-of-mercury or corrosive-sublimate process) for 

preserving timber 214 

Land and Land damages, cost 354 

Lateral bracing for trestles 141 

Length of rails 229 

a simple curve 20 

a spiral 53 

Level, dumpy, adjustments of — Appendix. p. 561 

wye, adjustments of — Appendix p. 539 

Leveling, location surveys, ,,,.,., 16 



776 INDEX. 

Level sections, volume of prismoids surveyed as 75 

numerical example , 76 

Life of locomotives , 327 

Limitations in location of track ^ _ , 34 

of maximum curvature , 429 

Lining of tunnels , 166 

Loading earthwork, cost 108 

of trestles I54 

Local traffic, definition and distinction from through 409 

Location of stations at distance from business centers, effect 376 

Location Surveys — paper location 15 

surveying methods. , 16 

Locomotives, as affected by changes in alignment, . , 406, 423, 437, 444, 450 

causes of deterioration , 406 

cost of renewals and repairs 389, 390 

^ general structure 316-326 

life of 327 

types permissible on sharp curvature 420, & 

(For details, look for the particular item.) 

Logarithmic sines and tangents of small angles — Table VI ,. p, 602 

sines, cosines, tangents, and cotangents — Table VII . . p. 605 

versed sines and external secants — Table VIII p. 650 

Logarithms of numbers — Table V p. 582 

Long chords for a 1° curve — Table II p. 568 

of a simple curve 21 

Longitudinal bracing of a trestle 140 

Longitudinals (rails) 201, 224 

Loop — see Spiral. 

Loosening earthwork, cost 107 

Loss in traffic due to lack of facilities 376 

Lundie's formula for train resistance 350 

Magnitude of railroad business. 363 

Maintenance of equipment, as affected by changes in curvature 423 

distance 406 

minor grades 437 

ruling grades 444 

piisher-engines 450 

cost of 388-392 

Maintenance of way as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher-engines 450 

cost of . . . 384-387 

Maps, use of, in reconnaissance 6 

Mass curve, area 102 

properties 101 

diagram, effect of change of grade line 104 

haul of earthwork 100 

value 103 

Mathematical design of switches 262-275 

Measurements, location surveys. 16 



INDEX. 777 

Mechanism of brakes , 335-337 

METAL TIES 218-223 

cost 222 

durability 219 

extent of use 218 

fastenings 221 

form and dimensions 220 

llfiddle areas, volume of prismoid computed from 73 

ordinate of a simple curve 21 

Mileage, car 402 

locomotives, average annual 327 

MINOR GRADES 435-439 

basis of cost 435 

classification 436 

effect on operating expenses 437 

estimate of cost of one foot of change of elevation. . . . 438 
operating value of the removal of a hump in a 

grade , 439 

Minor openings in road-bed 193-195 

Minor stations, rooms required, construction 287 

MISCELLANEOUS STRUCTURES AND BUILDINGS.— Chap. XII. 

Modifications in location, compound curves 39 

simple curves 33 

Mogul locomotives, running gear 323 

Monopoly, extent to which a railroad may be such 371 

Mountain routes — reconnoissance 5 

" Mud " baUast «... 197 

sills, trestle foundations. . 139, 6 

Multiform compound curves used as spirals 45 

Multiple story construction for trestles 137 

Myer's formula for culvert area , 180 

Natural sines, cosines, tangents, and cotangents — Table IX p. 695 

versed sines and external secants — Table X p. 718 

Non-competitive traffic, definition 409 

effect of variations in distance 413, 414 

extent of monopoly 371 

Notes — ^form for cross-sectioning 12 

location surveys 17 

reconnoissance 7 

Number of a frog, to find. 256 

of trains per day, probable 374 

Nut-locks, design = 253 

N. Y. Central formula for train resistance , 350 

Obstacles to location of trackwork 32 

Obstructed curve, in curve location 32, c 

Old lines, improvement of — Chap. XXIV 

rail culverts 190 

Open cuts vs. tunnels 174 

OPERATING EXPENSES.— Chap. XX. 

detailed classification — Table XX, pp. 470-473 
effect of change of grade — 439, and Table 

XXIV p. 520 



778 INDEX. 



operating expenses, effect of curvature on 421-426 

distance on 405-408 

estimated cost of each additional foot — Table 

XXI p. 48s 

of each additional mile — Table 

XXI p. 485 

of 1° additional of central angle 

—Table XXII p. 501 

fivefold distribution 379 

per train mile 380 

reasons for uniformity per train mile 381 

(For details look for the particular item.) 

Operation of trains, effect of curvature on 420 

Oscillatory and concussive velocity resistances, train 342 

Ordinates of a spiral 47 

Paper location in location surveys 15 

Passengers carried one mile « 363 

Physical tests of steel splice bars 243, a 

steel rails 233 

Picks, use inlloosening earth 107, b 

Pile bents 129, 133 

driving 130 

driving formulae 131 

points and shoes 132 

trestles, cost 134 

design 133 

PILE TRESTLES 129-134 

Pilot truck of locomotive, action 315 

PIPE CULVERTS 184-187 

advantages , 184 

construction 185 

iron. 186 

tile , 187 

Pipe compensator 309 

Pipes, use in block signaling. 309 

Pit cattle guards 193 

Platforms, station 286 

Ploughs, use in loosening earth 107, a 

Point of curve 21 

inaccessible, in curve location 32, 6 

Point of tangency 21 

inaccessible, in curve location, , 32, b 

Point-rails of switches, construction .0 258 

Point-switches 258 

Pony truck of locomotive, action 315 

Portals, tunnels, methods of excavation 173 

Posts, trestle, design of 157 

Preliminary financiering of railroads — Chap. XIX., and 352 

Preliminary surveys — cross-section method 11 

"first" and "second" . 14 

general character 10 



INDEX. 779 

Preliminary surveys, value of re-surveys at critical points 14 

Preservative processes for timber, cost 216 

general principle 2lf 

methods 211-215, 217 

Prismoidol correction for irregular prismoids, approximate value 84 

in earthwork computations, comparison of exact 

and approximate methods 85, 86 

formula, proof c . 71 

Prismoid, irregular, computation of volume 83 

Prismoids, in earthwork computations 67 

Profit and loss, dependence on business done 372 

small margin between them for railroad promoters 370 

Profits (and security) in the two general classes of railroad obligations. . . 369 

Profit, in earthwork operations 115 

PROMOTION OF RAILROAD PROJECTS.— Chap. XIX. 

Pumping, for locomotive water-tanks 283, 284 

Pusher grades 446-451 

comparative cost 450, 451 

general principles 446 

required balance between through and pusher grades . . 447 

required length 449 

Pusher engines, cost per mile — Table XXIX p. 540 

operation 448 

service o 450 

Radiation from locomotives 321 

into the exhaust-steam. 321 

Radii of curves^-Table I p. 5^4 

Radius of curvature (of track), relation to aperating expenses 421 

Rail braces 244 

expansion, resistance at joints and ties to free expansion 251 

FASTENINGS.— Chap. X. 

gap, effect of, at joints 239 

joints 237-243, a 

••Bonzano" 243 

••Cloud'' 243 

••Continuous" 243 

•*Fisher" 243 

••Weber" 24^ 

••100 per-cent" 243 

••bridge" 240 

effect of rail gap 239 

efficiency of angle-bar 238 

failures 241 

later designs 243 

specifications ,...,. 243, a 

••supported" 238,240 

"suspended". , 238,240 

theoretical requirements for perfect 237 

sections ., , . « . , 225, 226 

A.S.C.E „ .226 



780 INDEX. 

Rail sections, "bridge " 225 

"bull-headed" 225,226 

compouivi 240 

"pear'' section 225 

radius of upper corner, effect 2^6 

reversible 226 

"Stevens " 225 

"Vignoles" 225 

RAILS.— Chap. IX. 

branding 233, 11 

cast-iron 225 

cost 236, 358, c 

of, as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher engines 450 

of renewals of 234, 235, 384 

chemical composition 232, 233 

effect of stiffness on traction 228 

expansion 230 

stresses caused by prevention of expansion 230 

rules for allowing for 231 

inspection. 233-17 

length 229 

allowable variation 233, 8 

45- and 60-foot rails 229 

No. 2 233, 16 

relation of weight, strength, and stiffness 228 

temperature when exposed to sun 231 

testing 233 

tons per mile 358, c 

wear on curves ^ 235 

tangents 234 

weight, allowable variation 233-7 

for various kinds of traffic 358 

Rates based on distance, reasons 404 

through, method of division of 410 

Receipts (railroad), effect of distance on 409-416 

Reconnoissance over a cross-country route. 4 

surveying, leveling methods , 7 

surveys 1-9 

character of 1 

cross-country route 4 

distance measurements 8 

mountain route 5 

selection of general route 2 

value of high grade work 9 

through a river valley 3 

Reduction of barometer reading to 32" F. — Table XI p. 74' 

Reinf orced-concrete culverts 190, a 

ties 224, a 

Renewal of rails, cost of 234, 235, 384 



INDEX. 781 

Renewal of rails, cost of, as affected by changes in curvature 422 

distance 405 

minor grades 437 

, ruling grades 444 

pusher engines 450 

Renewal of tie* cost of 208, et seq. 384 

as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher engines 450 

regulations governing it 208 

Repairs and renewals of locomotives, cost 389, 390 

as affected by change of dis- 
tance 406 

curvature 423 

minor grades.. 437 
by ruling grades 444 

Repairs of roadway, cost of 385 

as affected by changes in curvature 422 

distance 405 

minor grades .... 437 
ruling grades .... 444 

pusher engines 450 

Repairs, wear, depreciation, and interest on cost of plant ; cost for earth- 
work operations 113 

Replacement of a compound curve by a curve with spirals 53 

simple curve by^ curve with spirals 51 

Requirements, nut-locks 251 

perfect rail-joint 237 

spikes 247 

track-bolts 251 

Resistances internal to the locomotive 341 

(see Train Resistance.) 

Revenue, gross, distribution of 378 

xioadbed, form of subgrade 63 

width for single and double track 62 

Roadway, cost of repairs of 385 

as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher engines 450 

Roadways, earthwork operations, cost of keeping in order 112 

Rock ballast o 197 

Rock cuts, compoimd sections 61, 62 

RolHng friction of wheels 343, a 

ROLLING STOCK.— Chap. XIV. 

Rotative kinetic energy of wheels of train 347, 431 

Rules for switch-laying 276 

Puling grades , 440-445 



782 INDEX. 

Ruling grades, choice of 441 

definition 3, 440 

operating value of a reduction in rate of 445 

proportion of traffic affected by 443 

Run-off for elevated outer rail 43 

Running gear of locomotives, types 323 

Sag in a grade, operating value of filling of 439 

Sand, used for ballast 197 

Scales, track 298 

Scrapers, use in earthwork 109, d 

Screws and bolts^ as rail-fastenings 249 

Section-houses, value, construction 288 

Selection of a general route for a railroad , 2 

Semaphore boards, in block signaling 308 

Setting tie-plates, methods 246 

Shafts, tunnel, design 167 

surveying 161 

Shells and small coal, used as ballast 197 

Shifting centers for locomotive pilot trucks, action. 315 

Shoveling (hand) of earthwork, cost 108, a 

(steam) of earthwork, cost 108, 6 

Shrinkage of earthwork , 96 

allowance 97 

Side-hill work, in earthwork computations 88 

correction for curvature 91 

Signaling, block, "absolute" blocking 304 

automatic , 305 

manual systems 302-304 

permissive 304 

Signals, mechanical details 308 

Sills for trestles, design 158 

Simple curves 18-36 

Skidding of wheels on rails 333, 334 

Slag, used for ballast 197 

Slide-rule, in earthwork computations 80 

Slipping of wheels on rails, lateral 312 

longitudinal, 311 

Slips, for switchwork 279. a 

Slopes in earthwork, for cut and fill 60, 62 

effect and value of sodding. 65 

Slope-stake rod, automatic 70 

Slope-stakes, determination of position 69 

Smith's formula for tram resistance 350 

Sodding slopes, effect and value 65 

Spacing of ties 206 

Span of trestles 138 

Specifications for earthwork 125 

steel rails. , 233 

steel splice-bars 243, a 

wooden ties 207 

Speed of trains, reduction due to curvature 419, a 

relation to superelevation of outer rail 41, 42 

relation to tractive adhesion 334, e 

Spikes, 247-250 



INDEX. 7S3 

Spikes, cost 358, c? 

driving , <, 248 

number per mile of track • 358, d 

requirements in design 247 

"wooden," for plugging spike-holes 250 

Spirals, bridge and tunnel 5 

(see Transition Curves.) 
Splice-bars — (see Angle-bars). 

Split stringers, caps, and sills 129; 143 

Sprague's formula for train resistance 350 

Spreading earthwork, cost Ill 

Stadia method — for preliminary surveys 13 

Stand pipes, for locomotive water-supply , . . . 285 

Starting grade at stations, reduction of 460 

Staybolts for locomotive fire-boxes 318 

Stays, in locomotive fire-box 318 

Steam pile-drivers 130 

Steam-shoveling of earthwork 108, b 

Stiffness of rails, effect on traction 228 

Stocks of railroads, security and profits 369 

Stone box culverts 189 

foundations for framed trestles 139, c 

Straight connecting curve between two parallel curved tracks 275, a 

from a curved main track 273 

frog rails, effect on design of switch 263 

point rails, effect on design of switch 264 

Strength of timber. 153 

factors of safety. 155 

required elements for trestles 152 

Stringer bridges, standard, steel 195 

Stringers, design 156 

for trestle floors 143 

Stub-switches 257 

Subchord, length. 19 

Subgrade, of roadbed, form , 63 

Superelevation of the outer rail on curves, L. V. R. R. run-off 43 

on trestles 147 

practical rules 42 

standard on N. Y. N. H. 

& H. R. R 42 

theory 41 

Superintendence, cost in earth operations 114 

of conducting transportation 394 

of maintenance of equipment 388 

Supported rail-joints 240 

Surface cattle guards 193, b 

surveys for tunneling 160 

Surveys and engineering expenses for railroads, cost 353 

accuracy 163 

for tunneling 160-163 

with compass 11 

Suspended rail-joints 240 



784 INDEX. 

Swinging pilot truck on locomotive 315 

Switchbacks 5 

Switch construction 254 261 

essential elements 254 

frogs 255, 256 

guard rails 261 

point 258 

stands 259 

stub 257 

tie rods 260 

SWITCHES AND CROSSINGS,. .Chap XI 

Switches, mathematical design 262-276 

comparison of methods 266 

using circular lead rails 262 

using straight frog rails 263 

using straight point rails 264 

using straight frog rails and straight 

point rails 265 

Switching engines, rimning gear 323 

used in pusher-engine service 448 

Switch leads and distances — Table III p. 57i 

laying, practical rules 276 

slips 276a 

stands ^^ 259 

TABLES. Numbers refer to pages, not sections. 

I. Radii of curves 564-567 

II. Tangents, external distances and long chords for a 1° curve, 

568-570 

III. Switch leads and distances 571, 572 

IV. Elements of transition curves 573-581 

V. Logarithms of numbers 582-601 

VI. Logarithmic sines and tangents of small angles 602-604 

VII. Logarithmic sines, cosines, tangents and cotangents ... . 605-649 

VIII. Logarithmic versed sines and external secants 650-694 

IX. Natural sines, cosines, tangents, and cotangents 695-717 

X. Natural versed sines and external secants , 718- 740 

XI. Reduction of barometer reading to 32° F 741 

XII. Barometric elevations 742 

XIII. Coeflficients for corrections for temperature and humidity. . . 742 

XIV. Capacity of cylindrical water-tanks in United States standard 

gallons of 231 cubic inches 329 

XV. Number of cross-ties per mile 430 

XVI. Tons per mile (with cost) of rails of various weights 431 

XVII. Splice bars for various weights of rails 432 

XVIII. Railroad spikes 433 

XIX. Track bolts, average number in a keg of 200 pounds 433 

XX. Classification of operating expenses of all railroads 470-473 

XXI. Effect on operating expenses of changes in distance 485 

XXII. Effect on operating expenses of changes in curvature 501 

XXIII. Velocity head of trains 612 

XXIV. Effect on operating expenses of 26.4 feet of rise and fall .... 520 
XXV. Tractive power of locomotivea ,.,.., , . . 525 



INDEX. 785 

TABLES. 

XXVI. Total train resistance per ton on various grades 527 

XXVII. Cost of an additional train to handle a given traffic 532 

XXVIII. Balanced grades for one, two, and three engines o . . . . 536 

XXIX. Cost for each mile of pusher-engine service 540 

XXX. Useful trigonometrical formulae 743, 744 

XXXI. Useful formulae and constants 745 

XXXII. Squares, cubes, squares root, cube roots, and reciprocals . .746-762 

XXXIII. Cubic yards per 100 feet of level sectior^ 763-765 

XXXIV. Annual charge against a tie, at 5% interest 766 

Talbot's formula for culvert area. . . . , 180 

Tamping for blasting 121 

Tangents for a 1° curve — Table II p. 568 

Tangent distance, simple curve 21 

Tanks, water, for locomotives 282 

capacity of cylindrical tanks. 282 

track 284 

Temperature allowances, while laying rails 231 

Ten-wheel locomotives, rimning gear 323 

Telegraph Unes for railroads, cost 361 

TERMINALS.— Chap. XIII. 

inconvenient, resulting loss 376 

justification for great expenditures 376 

Terminal pyramids and wedges, in earthwork 59 

Tests for splice bars 243, a 

for rails 233 

to measure the efficiency of brakes . 338 

Three-level sections in earthwork, determination of area 79 

numerical example. 79 

Throw of a switch , 262 

Through traffic, definition. , . . . , 409 

division of receipts between roads 410 

effect of changes in distance on receipts 411 

Tie-plates 244-246 

advantages 244 

elements of design 245 

method of setting 246 

Tie rods, for switches 260 

TIES.— Chap. VIII. 

cost of renewal of , , , 208 et seq. 384 

as affected by changes in curvature 422 

distance 405 

minor grades 437 

ruling grades 444 

pusher-engines 450 

metal , . , 218-223 

cost o 222 

durability 219 

extent of use , 218 

fastenings 221 

form and dimensions 220 

number per mile of track — Table XV p. 430 

on trestles 146 

wooden ,,..,...,.,,. , , 203-217 



786 INDEX. 

Ties, wooden, choice of wood 20^5 

construction, 207 

cost , , 209, 358, b 

dimensions „ , 205 

durability 204 

economics 202 

quality of timber 207 

spacing 206 

specifications 207 

Tile drains, to drain roadbed 64 

pipe culverts 187 

Timber, choice for trestles , , 149 

piles, o 129 

ties _ , 203 

strength of 153 

Tons carried one mile. 363 

Topographical maps, use of, in reconnaissance 6 

Track bolts, average number in a keg of 200 pounds.. 358, d 

cost 358, d 

design 252 

essential requirements. 251 

number required per mile 358, d 

circuit for automatic signaling 310 

laying on railroads, cost 358, e 

scales 299 

Tractive power of locomotives, Table XXV, p. 525 and 322 

relation to boiler and cylinder power. . 326 

Traffic, classification of ; . . . 409 

TRAIN-BRAKES 333-339 

automatic 337 

brake-shoes 339 

general principles 333, 334 

hand-brakes 335 

straight air-brakes 336 

tests for efficiency 338 

Train length limited by curvature 420, a 

load, financial value of increasing. 444 

maximum on any grade 442 

loads, methods of increasing 455, b, 458 et seq. 

RESISTANCE.— Chap. XVI. 

formulsp for 350 

total, per ton, on various grades — ^Table 

XXVI, p. 527, and 444 

Service, cost of, 399, and Table XX. 

as affected by changes in alignment, 407, 424, 437, 

444, 450 
supplies and expenses, cost of, in conducting transportation — 

400 and Table XX. 
wages — (see Train service). 

Transfer cranes in freight yards 298 

Transit, adjustments of • • • PP- 555-559 

Transition curves , 41-53 



INDEX. 787 

Transition curves, Table IV pp. 573-58i 

application to compound curves 52 

field work c 50 

fundamental principle 44 

replacing a compound curve by curves with spirals 53 

simple curve by a curve with spirals 51 

required length 46 

their relation to tangents and simple curves 49 

to find the deflections from any point 48 

ordinates 47 

use of Table IV 53, a 

Transportation, effect of curvature on conducting 424 

TRESTLES.— Chap. IV. 

cost 150 

extent of use 126 

framed 135-150 

pile 129-134 

posts, design 157 

required elements of strength 152 

sills, design 158 

stringers, design 156 

timber • 149, 153 

vs. embankments 127 

Trimming cuts to proper cross-section 112, a 

Trucks, car 330 

four-wheeled, action on curves 312 

locomotive pilot „ 315 

with shifting center .^ 315 

TUNNELS.— Chap. V. 

cost 175 

vs. open cuts 174 

Tunnel cross-sections 164 

design 164-168 

drains 168 

enlargement 170 

grade 165 

headings 169 

lining 166 

portals , 173 

shafts 161, 167 

spirals 5 

Turnout, connecting curve from a straight track 271 

from a curved track to the outside 272 

to the INSIDE 273 

double, from straight track 269 

dimensions, development of approximate rule for above. . . . 267 

from INNER side of curved track 268 

from OUTER side of curved track 267 

Turnouts with straight point rails and straight frog rails, dimensions of 

— Table III pp. 57i, 572 

two, on same side 270 

Turntables for locomotives 292 

Two-level ground, volume of prismoid surveyed as 74 



788 INDEX. 

Underground surveys in tunnels , 162 

Undulatory grades, advantages, disadvantages, and safe limits 434 

Unit chord, simple curves 18 

Upright switch-stands 259 

Useful formulae and constants — Table XXXI , p. 745 

trigonometrical formulae — Table XXX . pp. 743-744 

Valley route — reconnoissance 3 

Velocity head applied to theory of motion of trains 431 

as applied to determination of train resistance 349 

of trains — Table XXIII .p. 512 

Velocity of trains, method of obtaining 458 

resistances, train 342 

Ventilation of a tunnel during construction 172 

Vertex inaccessible, curve location 32, a 

of a curve 21 

Vertical curves, mathematical form 56 

necessity for use 54 

numerical example 57 

required length 55 

Virtual grade, reduction of 458-460 

profile, construction of •. 432 

use, value, and possible misuse 433 

Von Borrie's formula for train resistance 350 

Vulcanizing, for preserving wooden ties 211 

Wage of engine- and roundhouse-men 396; 407, item 80 

as affected by changes in align- 
ment 407, 424, 437, 444, 450, 

trackmen 405 

trainmen 399 

Wagons, use in hauling earthwork 109,6 

Water for locomotives, chemical qualities 281 

consumption and cost 398; 407, item 83 

methods of purification 281 

stations and water supply. 280-285 

\ location. . 280 

pumping. . 283 

required qualities of water 281 

stand-pipes 285 

tanks 282 

track tanks 284 

table in locomotive fire-box 318 

tanks for locomotives 282 

capacity of cylindrical tanks 282 

protection from freezing 282 

way for culverts 178-183 

Watering stock. . 369 

Wear of rails on curves 235 

on tangents 234 

Weight of rails, 226, 227, and Table XVI p. 43i 

Wellhouse (zinc-tannin) process for preserving timber 215 

Wellington's formula for train resistance 350 

Westinghouse air-brakes. 337 



INDEX. 789 

Wheelbarrows, use in hauling earthwork 109, c 

Wheel resistances, train 343 

Wheels and rails, mutual action and reaction 311-315 

effect of rigidly attaching them to axles 311 

White oak, use for trestles 129, 149 

ties 204 

Wire-drawn steam 321 

Wires and pipes, used in block signaling 309 

WolfiE's formula for train resistance 350 

Wooden box culverts 188 

spikes, for filling spike holes 250 

Yard-engine expenses 395 

YARDS AND TERMINALS.— Chap. XIII. 

Yards, engine 300 

freight 295-299 

grade in 295 

minor 297 

relation to main tracks 296 

transfer cranes 298 

track scales 299 

value of proper design 293 



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APR 8 1913 



